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| | | |
| {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | | {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" |
− | |+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math> | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
| |- style="height:48px; background:#f0f0ff" | | |- style="height:48px; background:#f0f0ff" |
| | <math>\text{Integer}\!</math> | | | <math>\text{Integer}\!</math> |
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| | | |
| {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" | | {| align="center" border="1" cellpadding="12" cellspacing="1" style="text-align:center; width:96%" |
− | |+ style="height:24px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math> | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
| |- style="height:48px; background:#f0f0ff" | | |- style="height:48px; background:#f0f0ff" |
| | <math>\text{Integer}\!</math> | | | <math>\text{Integer}\!</math> |
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| | | |
| {| align="center" border="1" width="96%" | | {| align="center" border="1" width="96%" |
− | |+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math> | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
| |- style="height:50px; background:#f0f0ff" | | |- style="height:50px; background:#f0f0ff" |
| | | | | |
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| | | |
| {| align="center" border="1" width="96%" | | {| align="center" border="1" width="96%" |
− | |+ style="height:25px" | <math>\text{Table 1.} ~~ \text{Prime Factorizations, Riffs, and Rotes}</math> | + | |+ style="height:24px" | <math>\text{Prime Factorizations, Riffs, Rotes, and Traversals}\!</math> |
| |- style="height:50px; background:#f0f0ff" | | |- style="height:50px; background:#f0f0ff" |
| | | | | |
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| & |P_k| | | & |P_k| |
| \\[10pt] | | \\[10pt] |
− | 0 & \{ 1 \} & 1 \\ | + | 0 & \{ 1 \} & 1 |
− | 1 & \{ 2 \} & 1 \\ | + | \\ |
− | 2 & \{ 3, 4 \} & 2 \\ | + | 1 & \{ 2 \} & 1 |
− | 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 \\ | + | \\ |
| + | 2 & \{ 3, 4 \} & 2 |
| + | \\ |
| + | 3 & \{ 5, 6, 7, 8, 9, 16 \} & 6 |
| + | \\ |
| 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 | | 4 & \{ 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536 \} & 20 |
| \end{array}</math> | | \end{array}</math> |
Line 1,098: |
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| \text{p}_3^1 | | \text{p}_3^1 |
| & = & \text{p}_{\text{p}_2^1}^1 | | & = & \text{p}_{\text{p}_2^1}^1 |
− | \\[10pt] | + | \\[12pt] |
| & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 | | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 |
| \end{array}</math> | | \end{array}</math> |
Line 1,120: |
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| \text{p}_4^1 | | \text{p}_4^1 |
| & = & \text{p}_{\text{p}_1^2}^1 | | & = & \text{p}_{\text{p}_1^2}^1 |
− | \\[10pt] | + | \\[12pt] |
| & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 | | & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 |
| \end{array}</math> | | \end{array}</math> |
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| \text{p}_1^3 | | \text{p}_1^3 |
| & = & \text{p}_1^{\text{p}_2^1} | | & = & \text{p}_1^{\text{p}_2^1} |
− | \\[10pt] | + | \\[12pt] |
| & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} | | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^1} |
| \end{array}</math> | | \end{array}</math> |
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| \text{p}_1^4 | | \text{p}_1^4 |
| & = & \text{p}_1^{\text{p}_1^2} | | & = & \text{p}_1^{\text{p}_1^2} |
− | \\[10pt] | + | \\[12pt] |
| & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} | | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^1}} |
| \end{array}</math> | | \end{array}</math> |
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| | [[Image:Rote 16 Big.jpg|90px]] | | | [[Image:Rote 16 Big.jpg|90px]] |
| |} | | |} |
− | |}
| |
− |
| |
− | ===ASCII===
| |
− |
| |
− | <pre>
| |
− | Example
| |
− |
| |
− | * k | natural numbers n such that |riff(n)| = k
| |
− | * 0 | 1;
| |
− | * 1 | 2;
| |
− | * 2 | 3, 4;
| |
− | * 3 | 5, 6, 7, 8, 9, 16;
| |
− | * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536;
| |
− | * The natural number values for the riffs with at most 3 pts are as follows (x = root):
| |
− | * .................o.......o..o.......o
| |
− | * .................|.......^..|.......^
| |
− | * .................v.......|..v.......|
| |
− | * ...........o..o..o....o..o..o..o.o..o
| |
− | * ...........|..^..|....|..|..^..|.^..^
| |
− | * ...........v..|..v....v..v..|..v/...|
| |
− | * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x;
| |
− | * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16;
| |
− | </pre>
| |
− |
| |
− | ==A062537==
| |
− |
| |
− | * [http://oeis.org/wiki/A062537 A062537]
| |
− |
| |
− | ===Wiki + TeX + JPEG===
| |
− |
| |
− | {| align="center" border="1" cellpadding="6"
| |
− | |+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math>
| |
− | | valign="bottom" |
| |
− | <p> </p><br>
| |
− | <p> </p><br>
| |
− | <p><math>a(1) ~=~ 0</math></p>
| |
− | | valign="bottom" |
| |
− | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br>
| |
− | <p><math>\text{p}\!</math></p><br>
| |
− | <p><math>a(2) ~=~ 1</math></p>
| |
− | | valign="bottom" |
| |
− | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br>
| |
− | <p><math>\text{p}_\text{p}\!</math></p><br>
| |
− | <p><math>a(3) ~=~ 2</math></p>
| |
− | | valign="bottom" |
| |
− | <p>[[Image:Riff 4 Big.jpg|40px]]</p><br>
| |
− | <p><math>\text{p}^\text{p}\!</math></p><br>
| |
− | <p><math>a(4) ~=~ 2</math></p>
| |
− | | valign="bottom" |
| |
− | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br>
| |
− | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
| |
− | <p><math>a(5) ~=~ 3</math></p>
| |
| |- | | |- |
− | | valign="bottom" | | + | | |
− | <p>[[Image:Riff 6 Big.jpg|65px]]</p><br> | + | {| cellpadding="12" style="text-align:center; width:100%" |
− | <p><math>\text{p} \text{p}_{\text{p}}\!</math></p><br> | + | | width="10%" | <math>10\!</math> |
− | <p><math>a(6) ~=~ 3</math></p>
| + | | width="25%" | |
− | | valign="bottom" | | + | <math>\begin{array}{lll} |
− | <p>[[Image:Riff 7 Big.jpg|65px]]</p><br>
| + | \text{p}_1^1 \text{p}_3^1 |
− | <p><math>\text{p}_{\text{p}^{\text{p}}}\!</math></p><br>
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_2^1}^1 |
− | <p><math>a(7) ~=~ 3</math></p>
| + | \\[12pt] |
− | | valign="bottom" | | + | & = & \text{p}_1^1 \text{p}_{\text{p}_{\text{p}_1^1}^1}^1 |
− | <p>[[Image:Riff 8 Big.jpg|65px]]</p><br>
| + | \end{array}</math> |
− | <p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br>
| + | | width="15%" | <math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math> |
− | <p><math>a(8) ~=~ 3</math></p>
| + | | width="25%" | [[Image:Riff 10 Big.jpg|90px]] |
− | | valign="bottom" | | + | | width="25%" | [[Image:Rote 10 Big.jpg|80px]] |
− | <p>[[Image:Riff 9 Big.jpg|40px]]</p><br>
| + | |- |
− | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> | + | | <math>11\!</math> |
− | <p><math>a(9) ~=~ 3</math></p>
| + | | |
− | | valign="bottom" | | + | <math>\begin{array}{lll} |
− | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br>
| + | \text{p}_5^1 |
− | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
| + | & = & \text{p}_{\text{p}_3^1}^1 |
− | <p><math>a(10) ~=~ 4</math></p>
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_{\text{p}_2^1}^1}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1}^1 |
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math> |
| + | | [[Image:Riff 11 Big.jpg|90px]] |
| + | | [[Image:Rote 11 Big.jpg|40px]] |
| + | |- |
| + | | <math>12\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_1^2 \text{p}_2^1 |
| + | & = & \text{p}_1^{\text{p}_1^1} \text{p}_{\text{p}_1^1}^1 |
| + | \end{array}</math> |
| + | | <math>\text{p}^{\text{p}} \text{p}_{\text{p}}\!</math> |
| + | | [[Image:Riff 12 Big.jpg|65px]] |
| + | | [[Image:Rote 12 Big.jpg|105px]] |
| + | |- |
| + | | <math>13\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_6^1 |
| + | & = & \text{p}_{\text{p}_1^1 \text{p}_2^1}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1}^1 |
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math> |
| + | | [[Image:Riff 13 Big.jpg|65px]] |
| + | | [[Image:Rote 13 Big.jpg|80px]] |
| + | |- |
| + | | <math>14\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_1^1 \text{p}_4^1 |
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^2}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^{\text{p}_1^1}}^1 |
| + | \end{array}</math> |
| + | | <math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math> |
| + | | [[Image:Riff 14 Big.jpg|90px]] |
| + | | [[Image:Rote 14 Big.jpg|105px]] |
| + | |- |
| + | | <math>17\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_7^1 |
| + | & = & \text{p}_{\text{p}_4^1}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_{\text{p}_1^2}^1}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1}^1 |
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math> |
| + | | [[Image:Riff 17 Big.jpg|90px]] |
| + | | [[Image:Rote 17 Big.jpg|65px]] |
| + | |- |
| + | | <math>18\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_1^1 \text{p}_2^2 |
| + | & = & \text{p}_1^1 \text{p}_{\text{p}_1^1}^{\text{p}_1^1} |
| + | \end{array}</math> |
| + | | <math>\text{p} \text{p}_{\text{p}}^{\text{p}}\!</math> |
| + | | [[Image:Riff 18 Big.jpg|65px]] |
| + | | [[Image:Rote 18 Big.jpg|120px]] |
| + | |- |
| + | | <math>19\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_8^1 |
| + | & = & \text{p}_{\text{p}_1^3}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_1^{\text{p}_2^1}}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}}^1 |
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math> |
| + | | [[Image:Riff 19 Big.jpg|90px]] |
| + | | [[Image:Rote 19 Big.jpg|65px]] |
| |- | | |- |
− | | valign="bottom" | | + | | <math>23\!</math> |
− | <p>[[Image:Riff 11 Big.jpg|90px]]</p><br> | + | | |
− | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br>
| + | <math>\begin{array}{lll} |
− | <p><math>a(11) ~=~ 4</math></p>
| + | \text{p}_9^1 |
− | | valign="bottom" | | + | & = & \text{p}_{\text{p}_2^2}^1 |
− | <p>[[Image:Riff 12 Big.jpg|65px]]</p><br>
| + | \\[12pt] |
− | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br>
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}}^1 |
− | <p><math>a(12) ~=~ 4</math></p>
| + | \end{array}</math> |
− | | valign="bottom" | | + | | <math>\text{p}_{\text{p}_{\text{p}}^{\text{p}}}\!</math> |
− | <p>[[Image:Riff 13 Big.jpg|65px]]</p><br>
| + | | [[Image:Riff 23 Big.jpg|65px]] |
− | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br>
| + | | [[Image:Rote 23 Big.jpg|80px]] |
− | <p><math>a(13) ~=~ 4</math></p>
| + | |- |
− | | valign="bottom" |
| + | | <math>25\!</math> |
− | <p>[[Image:Riff 14 Big.jpg|90px]]</p><br>
| + | | |
− | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br>
| + | <math>\begin{array}{lll} |
− | <p><math>a(14) ~=~ 4</math></p>
| + | \text{p}_3^2 |
− | | valign="bottom" | | + | & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} |
− | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br>
| + | \\[12pt] |
− | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
| + | & = & \text{p}_{\text{p}_{\text{p}_1^1}^1}^{\text{p}_1^1} |
− | <p><math>a(15) ~=~ 5</math></p>
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math> |
| + | | [[Image:Riff 25 Big.jpg|65px]] |
| + | | [[Image:Rote 25 Big.jpg|80px]] |
| |- | | |- |
− | | valign="bottom" | | + | | <math>27\!</math> |
− | <p>[[Image:Riff 16 Big.jpg|65px]]</p><br> | + | | |
− | <p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br>
| + | <math>\begin{array}{lll} |
− | <p><math>a(16) ~=~ 3</math></p>
| + | \text{p}_2^3 |
− | | valign="bottom" |
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} |
− | <p>[[Image:Riff 17 Big.jpg|90px]]</p><br>
| + | \\[12pt] |
− | <p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br>
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_{\text{p}_1^1}^1} |
− | <p><math>a(17) ~=~ 4</math></p>
| + | \end{array}</math> |
− | | valign="bottom" |
| + | | <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math> |
− | <p>[[Image:Riff 18 Big.jpg|65px]]</p><br>
| + | | [[Image:Riff 27 Big.jpg|65px]] |
− | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br>
| + | | [[Image:Rote 27 Big.jpg|80px]] |
− | <p><math>a(18) ~=~ 4</math></p>
| + | |- |
− | | valign="bottom" |
| + | | <math>32\!</math> |
− | <p>[[Image:Riff 19 Big.jpg|90px]]</p><br>
| |
− | <p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br>
| |
− | <p><math>a(19) ~=~ 4</math></p>
| |
− | | valign="bottom" | | |
− | <p>[[Image:Riff 20 Big.jpg|90px]]</p><br>
| |
− | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br>
| |
− | <p><math>a(20) ~=~ 5</math></p>
| |
− | |}
| |
− | | |
− | ==A109300==
| |
− | | |
− | * [http://oeis.org/wiki/A109300 A109300]
| |
− | | |
− | ===JPEG===
| |
− | | |
− | {| align="center" border="1" cellpadding="10"
| |
| | | | | |
− | <p>[[Image:Rote 3 Big.jpg|40px]]</p><br> | + | <math>\begin{array}{lll} |
− | <p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p>
| + | \text{p}_1^5 |
| + | & = & \text{p}_1^{\text{p}_3^1} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_{\text{p}_2^1}^1} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_{\text{p}_{\text{p}_1^1}^1}^1} |
| + | \end{array}</math> |
| + | | <math>\text{p}^{\text{p}_{\text{p}_{\text{p}}}}\!</math> |
| + | | [[Image:Riff 32 Big.jpg|90px]] |
| + | | [[Image:Rote 32 Big.jpg|65px]] |
| + | |- |
| + | | <math>49\!</math> |
| | | | | |
− | <p>[[Image:Rote 4 Big.jpg|65px]]</p><br> | + | <math>\begin{array}{lll} |
− | <p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p>
| + | \text{p}_4^2 |
| + | & = & \text{p}_{\text{p}_1^2}^{\text{p}_1^1} |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^1}}^{\text{p}_1^1} |
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}^{\text{p}}}^{\text{p}}\!</math> |
| + | | [[Image:Riff 49 Big.jpg|65px]] |
| + | | [[Image:Rote 49 Big.jpg|80px]] |
| + | |- |
| + | | <math>53\!</math> |
| | | | | |
− | <p>[[Image:Rote 6 Big.jpg|80px]]</p><br>
| + | <math>\begin{array}{lll} |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p>
| + | \text{p}_{16}^1 |
| + | & = & \text{p}_{\text{p}_1^4}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^2}}^1 |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}}^1 |
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}^{\text{p}^{\text{p}}}}\!</math> |
| + | | [[Image:Riff 53 Big.jpg|90px]] |
| + | | [[Image:Rote 53 Big.jpg|90px]] |
| + | |- |
| + | | <math>64\!</math> |
| | | | | |
− | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> | + | <math>\begin{array}{lll} |
− | <p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p>
| + | \text{p}_1^6 |
| + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_2^1} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_1^1 \text{p}_{\text{p}_1^1}^1} |
| + | \end{array}</math> |
| + | | <math>\text{p}^{\text{p} \text{p}_{\text{p}}}\!</math> |
| + | | [[Image:Riff 64 Big.jpg|65px]] |
| + | | [[Image:Rote 64 Big.jpg|105px]] |
| + | |- |
| + | | <math>81\!</math> |
| | | | | |
− | <p>[[Image:Rote 12 Big.jpg|105px]]</p><br>
| + | <math>\begin{array}{lll} |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p>
| + | \text{p}_2^4 |
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^2} |
| + | \\[12pt] |
| + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_1^{\text{p}_1^1}} |
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}}^{\text{p}^{\text{p}}}\!</math> |
| + | | [[Image:Riff 81 Big.jpg|65px]] |
| + | | [[Image:Rote 81 Big.jpg|105px]] |
| + | |- |
| + | | <math>128\!</math> |
| | | | | |
− | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> | + | <math>\begin{array}{lll} |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p>
| + | \text{p}_1^7 |
| + | & = & \text{p}_1^{\text{p}_4^1} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^2}^1} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^{\text{p}_1^1}}^1} |
| + | \end{array}</math> |
| + | | <math>\text{p}^{\text{p}_{\text{p}^{\text{p}}}}\!</math> |
| + | | [[Image:Riff 128 Big.jpg|90px]] |
| + | | [[Image:Rote 128 Big.jpg|90px]] |
| + | |- |
| + | | <math>256\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_1^8 |
| + | & = & \text{p}_1^{\text{p}_1^3} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_2^1}} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_{\text{p}_1^1}^1}} |
| + | \end{array}</math> |
| + | | <math>\text{p}^{\text{p}^{\text{p}_{\text{p}}}}\!</math> |
| + | | [[Image:Riff 256 Big.jpg|90px]] |
| + | | [[Image:Rote 256 Big.jpg|90px]] |
| + | |- |
| + | | <math>512\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \text{p}_1^9 |
| + | & = & \text{p}_1^{\text{p}_2^2} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_{\text{p}_1^1}^{\text{p}_1^1}} |
| + | \end{array}</math> |
| + | | <math>\text{p}^{\text{p}_{\text{p}}^{\text{p}}}\!</math> |
| + | | [[Image:Riff 512 Big.jpg|65px]] |
| + | | [[Image:Rote 512 Big.jpg|105px]] |
| + | |- |
| + | | <math>65536\!</math> |
| | | | | |
− | <p>[[Image:Rote 36 Big.jpg|145px]]</p><br> | + | <math>\begin{array}{lll} |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p>
| + | \text{p}_1^{16} |
| + | & = & \text{p}_1^{\text{p}_1^4} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^2}} |
| + | \\[12pt] |
| + | & = & \text{p}_1^{\text{p}_1^{\text{p}_1^{\text{p}_1^1}}} |
| + | \end{array}</math> |
| + | | <math>\text{p}^{\text{p}^{\text{p}^{\text{p}}}}\!</math> |
| + | | [[Image:Riff 65536 Big.jpg|90px]] |
| + | | [[Image:Rote 65536 Big.jpg|115px]] |
| + | |} |
| |} | | |} |
| | | |
Line 1,313: |
Line 1,427: |
| Example | | Example |
| | | |
− | * Table of Rotes and Primal Functions for Positive Integers of Rote Height 2 | + | * k | natural numbers n such that |riff(n)| = k |
− | * | + | * 0 | 1; |
− | * o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o | + | * 1 | 2; |
− | * | | | | | | | | | | | | | + | * 2 | 3, 4; |
− | * o-o o-o o-o o-o o---o o-o o-o o-o o---o o-o o---o | + | * 3 | 5, 6, 7, 8, 9, 16; |
− | * | | | | | | | | | | | | + | * 4 | 10, 11, 12, 13, 14, 17, 18, 19, 23, 25, 27, 32, 49, 53, 64, 81, 128, 256, 512, 65536; |
− | * O O O===O O O=====O O===O O=====O | + | * The natural number values for the riffs with at most 3 pts are as follows (x = root): |
− | *
| + | * .................o.......o..o.......o |
− | * 2:1 1:2 1:1 2:1 2:2 1:2 2:1 1:1 2:2 1:2 2:2 | + | * .................|.......^..|.......^ |
− | *
| + | * .................v.......|..v.......| |
− | * 3 4 6 9 12 18 36
| + | * ...........o..o..o....o..o..o..o.o..o |
− | *
| + | * ...........|..^..|....|..|..^..|.^..^ |
| + | * ...........v..|..v....v..v..|..v/...| |
| + | * Riff:...x;.x,.x;.x,.x.x,.x,.x,.x,...x; |
| + | * Value:..2;.3,.4;.5,..6.,.7,.8,.9,..16; |
| </pre> | | </pre> |
| | | |
− | ==A109301== | + | ==A062537== |
| | | |
− | * [http://oeis.org/wiki/A109301 A109301] | + | * [http://oeis.org/wiki/A062537 A062537] |
| | | |
− | ===JPEG=== | + | ===Wiki + TeX + JPEG=== |
| | | |
− | {| align="center" border="1" cellpadding="6" | + | {| align="center" border="1" cellpadding="10" |
| + | |+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rooted Node Big.jpg|20px]]</p><br> | + | <p> </p><br> |
− | <p><math>\begin{array}{l} \varnothing \\ 1 \end{array}</math></p> | + | <p><math>1\!</math></p><br> |
| + | <p><math>a(1) ~=~ 0</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rote 2 Big.jpg|40px]]</p><br> | + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 \\ 2 \end{array}</math></p> | + | <p><math>\text{p}\!</math></p><br> |
| + | <p><math>a(2) ~=~ 1</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rote 3 Big.jpg|40px]]</p><br> | + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p> | + | <p><math>\text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(3) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rote 4 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 4 Big.jpg|40px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p> | + | <p><math>\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(4) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rote 5 Big.jpg|40px]]</p><br> | + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 3\!:\!1 \\ 5 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| + | <p><math>a(5) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rote 6 Big.jpg|80px]]</p><br> | + | <p>[[Image:Riff 6 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}}\!</math></p><br> |
| + | <p><math>a(6) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rote 7 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 7 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 4\!:\!1 \\ 7 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}^{\text{p}}}\!</math></p><br> |
| + | <p><math>a(7) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Rote 8 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 8 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!3 \\ 8 \end{array}</math></p> | + | <p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br> |
− | | valign="bottom" | | + | <p><math>a(8) ~=~ 3</math></p> |
− | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> | + | | valign="bottom" | |
− | <p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p> | + | <p>[[Image:Riff 9 Big.jpg|40px]]</p><br> |
− | | valign="bottom" | | + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> |
− | <p>[[Image:Rote 10 Big.jpg|80px]]</p><br> | + | <p><math>a(9) ~=~ 3</math></p> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}</math></p> | + | | valign="bottom" | |
− | |- | + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> |
− | | valign="bottom" | | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | <p>[[Image:Rote 11 Big.jpg|40px]]</p><br> | + | <p><math>a(10) ~=~ 4</math></p> |
− | <p><math>\begin{array}{l} 5\!:\!1 \\ 11 \end{array}</math></p> | + | |- |
− | | valign="bottom" | | + | | valign="bottom" | |
− | <p>[[Image:Rote 12 Big.jpg|105px]]</p><br> | + | <p>[[Image:Riff 11 Big.jpg|90px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br> |
− | | valign="bottom" | | + | <p><math>a(11) ~=~ 4</math></p> |
− | <p>[[Image:Rote 13 Big.jpg|80px]]</p><br> | + | | valign="bottom" | |
− | <p><math>\begin{array}{l} 6\!:\!1 \\ 13 \end{array}</math></p> | + | <p>[[Image:Riff 12 Big.jpg|65px]]</p><br> |
− | | valign="bottom" | | + | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> |
− | <p>[[Image:Rote 14 Big.jpg|105px]]</p><br> | + | <p><math>a(12) ~=~ 4</math></p> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}</math></p> | + | | valign="bottom" | |
− | | valign="bottom" | | + | <p>[[Image:Riff 13 Big.jpg|65px]]</p><br> |
− | <p>[[Image:Rote 15 Big.jpg|80px]]</p><br> | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}</math></p> | + | <p><math>a(13) ~=~ 4</math></p> |
− | |- | + | | valign="bottom" | |
− | | valign="bottom" | | + | <p>[[Image:Riff 14 Big.jpg|90px]]</p><br> |
− | <p>[[Image:Rote 16 Big.jpg|90px]]</p><br> | + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br> |
− | <p><math>\begin{array}{l} 1\!:\!4 \\ 16 \end{array}</math></p> | + | <p><math>a(14) ~=~ 4</math></p> |
− | | valign="bottom" | | + | | valign="bottom" | |
− | <p>[[Image:Rote 17 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> |
− | <p><math>\begin{array}{l} 7\!:\!1 \\ 17 \end{array}</math></p> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | | valign="bottom" | | + | <p><math>a(15) ~=~ 5</math></p> |
− | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> | + | |- |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p> | + | | valign="bottom" | |
− | | valign="bottom" | | + | <p>[[Image:Riff 16 Big.jpg|65px]]</p><br> |
− | <p>[[Image:Rote 19 Big.jpg|65px]]</p><br> | + | <p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br> |
− | <p><math>\begin{array}{l} 8\!:\!1 \\ 19 \end{array}</math></p> | + | <p><math>a(16) ~=~ 3</math></p> |
− | | valign="bottom" | | + | | valign="bottom" | |
− | <p>[[Image:Rote 20 Big.jpg|105px]]</p><br> | + | <p>[[Image:Riff 17 Big.jpg|90px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br> |
− | |- | + | <p><math>a(17) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 18 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(18) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 19 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br> |
| + | <p><math>a(19) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 20 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| + | <p><math>a(20) ~=~ 5</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 21 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(21) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 22 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(22) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 23 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(23) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 24 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(24) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 25 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(25) ~=~ 4</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 26 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(26) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 27 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(27) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 28 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(28) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 29 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(29) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(30) ~=~ 6</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 31 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br> |
| + | <p><math>a(31) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 32 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(32) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 33 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(33) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 34 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(34) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 35 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(35) ~=~ 6</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 36 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(36) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 37 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(37) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 38 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(38) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 39 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(39) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 40 Big.jpg|135px]]</p><br> |
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(40) ~=~ 6</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 41 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(41) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 42 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(42) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 43 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(43) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 44 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(44) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 45 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(45) ~=~ 6</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 46 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(46) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 47 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(47) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 48 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(48) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 49 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(49) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 50 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(50) ~=~ 5</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 51 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(51) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 52 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(52) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 53 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(53) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 54 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(54) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(55) ~=~ 7</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 56 Big.jpg|135px]]</p><br> |
| + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(56) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 57 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(57) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 58 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(58) ~=~ 6</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 59 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br> |
| + | <p><math>a(59) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 60 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(60) ~=~ 7</math></p> |
| + | |} |
| + | |
| + | ==A062860== |
| + | |
| + | * [http://oeis.org/wiki/A062860 A062860] |
| + | |
| + | ===Wiki + TeX + JPEG=== |
| + | |
| + | {| align="center" border="1" cellpadding="10" |
| + | |+ style="height:25px" | <math>a(n) = \text{Least Integer}~ j ~\text{with}~ n ~\text{Nodes in Its Riff}</math> |
| + | | valign="bottom" | |
| + | <p> </p><br> |
| + | <p><math>1\!</math></p><br> |
| + | <p><math>a(0) ~=~ 1</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> |
| + | <p><math>\text{p}\!</math></p><br> |
| + | <p><math>a(1) ~=~ 2</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> |
| + | <p><math>\text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(2) ~=~ 3</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| + | <p><math>a(3) ~=~ 5</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| + | <p><math>a(4) ~=~ 10</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
| + | <p><math>a(5) ~=~ 15</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(6) ~=~ 30</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(7) ~=~ 55</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 105 Big.jpg|115px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(8) ~=~ 105</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Riff 165 Big.jpg|135px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(9) ~=~ 165</math></p> |
| + | |} |
| + | |
| + | ==A106177== |
| + | |
| + | * [http://oeis.org/wiki/A106177 A106177] |
| + | |
| + | ===Primal Codes of Finite Partial Functions on Positive Integers=== |
| + | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{array}{rcl} |
| + | 1 & = & \varnothing \\ |
| + | 2 & = & 1\!:\!1 \\ |
| + | 3 & = & 2\!:\!1 \\ |
| + | 4 & = & 1\!:\!2 \\ |
| + | 5 & = & 3\!:\!1 \\ |
| + | 6 & = & 1\!:\!1 ~~ 2\!:\!1 \\ |
| + | 7 & = & 4\!:\!1 \\ |
| + | 8 & = & 1\!:\!3 \\ |
| + | 9 & = & 2\!:\!2 \\ |
| + | 10 & = & 1\!:\!1 ~~ 3\!:\!1 \\ |
| + | 11 & = & 5\!:\!1 \\ |
| + | 12 & = & 1\!:\!2 ~~ 2\!:\!1 \\ |
| + | 13 & = & 6\!:\!1 \\ |
| + | 14 & = & 1\!:\!1 ~~ 4\!:\!1 \\ |
| + | 15 & = & 2\!:\!1 ~~ 3\!:\!1 \\ |
| + | 16 & = & 1\!:\!4 \\ |
| + | 17 & = & 7\!:\!1 \\ |
| + | 18 & = & 1\!:\!1 ~~ 2\!:\!2 \\ |
| + | 19 & = & 8\!:\!1 \\ |
| + | 20 & = & 1\!:\!2 ~~ 3\!:\!1 |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | ===Wiki Table=== |
| + | |
| + | {| align="center" style="font-weight:bold; text-align:center" |
| + | | || || || || || || || || |
| + | | <font color="red">1</font> |
| + | | |
| + | | <font color="red">1</font> |
| + | |- |
| + | | || || || || || || || |
| + | | <font color="red">2</font> |
| + | | || 1 || |
| + | | <font color="red">2</font> |
| + | |- |
| + | | || || || || || || |
| + | | <font color="red">3</font> |
| + | | || 1 || || 1 || |
| + | | <font color="red">3</font> |
| + | |- |
| + | | || || || || || |
| + | | <font color="red">4</font> |
| + | | || 1 || || 2 || || 1 || |
| + | | <font color="red">4</font> |
| + | |- |
| + | | || || || || |
| + | | <font color="red">5</font> |
| + | | || 1 || || 3 || || 1 || || 1 || |
| + | | <font color="red">5</font> |
| + | |- |
| + | | || || || |
| + | | <font color="red">6</font> |
| + | | || 1 || || 1 || || 1 || || 4 || || 1 || |
| + | | <font color="red">6</font> |
| + | |- |
| + | | || || |
| + | | <font color="red">7</font> |
| + | | || 1 || || 5 || || 2 || || 9 || || 1 || || 1 || |
| + | | <font color="red">7</font> |
| + | |- |
| + | | || |
| + | | <font color="red">8</font> |
| + | | || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">8</font> |
| + | |- |
| + | | |
| + | | <font color="red">9</font> |
| + | | || 1 || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || 1 || |
| + | | <font color="red">9</font> |
| + | |- |
| + | | width="12pt" | <font color="red">10</font> |
| + | | width="12pt" | |
| + | | width="12pt" | 1 |
| + | | width="12pt" | |
| + | | width="12pt" | 1 |
| + | | width="12pt" | |
| + | | width="12pt" | 1 |
| + | | width="12pt" | |
| + | | width="12pt" | 36 |
| + | | width="12pt" | |
| + | | width="12pt" | 1 |
| + | | width="12pt" | |
| + | | width="12pt" | 2 |
| + | | width="12pt" | |
| + | | width="12pt" | 1 |
| + | | width="12pt" | |
| + | | width="12pt" | 8 |
| + | | width="12pt" | |
| + | | width="12pt" | 1 |
| + | | width="12pt" | |
| + | | width="12pt" | <font color="red">10</font> |
| + | |} |
| + | |
| + | ===Wiki + TeX=== |
| + | |
| + | ====Smallmatrix==== |
| + | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | & & & & & & & & & {\color{red}1} & & {\color{red}1} |
| + | \\ |
| + | & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} |
| + | \\ |
| + | & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} |
| + | \\ |
| + | & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} |
| + | \\ |
| + | & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} |
| + | \\ |
| + | & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} |
| + | \\ |
| + | & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} |
| + | \\ |
| + | & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} |
| + | \\ |
| + | & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} |
| + | \\ |
| + | {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | |
| + | ====Array==== |
| + | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{array}{*{21}{c}} |
| + | & & & & & & & & & {\color{red}1} & & {\color{red}1} |
| + | \\ |
| + | & & & & & & & & {\color{red}2} & & 1 & & {\color{red}2} |
| + | \\ |
| + | & & & & & & & {\color{red}3} & & 1 & & 1 & & {\color{red}3} |
| + | \\ |
| + | & & & & & & {\color{red}4} & & 1 & & 2 & & 1 & & {\color{red}4} |
| + | \\ |
| + | & & & & & {\color{red}5} & & 1 & & 3 & & 1 & & 1 & & {\color{red}5} |
| + | \\ |
| + | & & & & {\color{red}6} & & 1 & & 1 & & 1 & & 4 & & 1 & & {\color{red}6} |
| + | \\ |
| + | & & & {\color{red}7} & & 1 & & 5 & & 2 & & 9 & & 1 & & 1 & & {\color{red}7} |
| + | \\ |
| + | & & {\color{red}8} & & 1 & & 6 & & 1 & & 1 & & 1 & & 2 & & 1 & & {\color{red}8} |
| + | \\ |
| + | & {\color{red}9} & & 1 & & 7 & & 1 & & 25 & & 1 & & 3 & & 1 & & 1 & & {\color{red}9} |
| + | \\ |
| + | {\color{red}10} & & 1 & & 1 & & 1 & & 36 & & 1 & & 2 & & 1 & & 8 & & 1 & & {\color{red}10} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | ====Matrix==== |
| + | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{matrix} |
| + | n \circ m |
| + | \\ |
| + | 1 ~/~\backslash~ 1 |
| + | \\ |
| + | 2 ~/~ 1 ~\backslash~ 2 |
| + | \\ |
| + | 3 ~/~ 1 \cdot 1 ~\backslash~ 3 |
| + | \\ |
| + | 4 ~/~ 1 \cdot 2 \cdot 1 ~\backslash~ 4 |
| + | \\ |
| + | 5 ~/~ 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 5 |
| + | \\ |
| + | 6 ~/~ 1 \cdot 1 \cdot 1 \cdot 4 \cdot 1 ~\backslash~ 6 |
| + | \\ |
| + | 7 ~/~ 1 \cdot 5 \cdot 2 \cdot 9 \cdot 1 \cdot 1 ~\backslash~ 7 |
| + | \\ |
| + | 8 ~/~ 1 \cdot 6 \cdot 1 \cdot 1 \cdot 1 \cdot 2 \cdot 1 ~\backslash~ 8 |
| + | \\ |
| + | 9 ~/~ 1 \cdot 7 \cdot 1 \cdot 25\cdot 1 \cdot 3 \cdot 1 \cdot 1 ~\backslash~ 9 |
| + | \\ |
| + | 10 ~/~ 1 \cdot 1 \cdot 1 \cdot 36\cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 1 ~\backslash~ 10 |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | ===ASCII=== |
| + | |
| + | <pre> |
| + | Example |
| + | |
| + | * n o m |
| + | * \ / |
| + | * 1 . 1 |
| + | * \ / \ / |
| + | * 2 . 1 . 2 |
| + | * \ / \ / \ / |
| + | * 3 . 1 . 1 . 3 |
| + | * \ / \ / \ / \ / |
| + | * 4 . 1 . 2 . 1 . 4 |
| + | * \ / \ / \ / \ / \ / |
| + | * 5 . 1 . 3 . 1 . 1 . 5 |
| + | * \ / \ / \ / \ / \ / \ / |
| + | * 6 . 1 . 1 . 1 . 4 . 1 . 6 |
| + | * \ / \ / \ / \ / \ / \ / \ / |
| + | * 7 . 1 . 5 . 2 . 9 . 1 . 1 . 7 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 8 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 9 . 1 . 7 . 1 . 25. 1 . 3 . 1 . 1 . 9 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 10 . 1 . 1 . 1 . 36. 1 . 2 . 1 . 8 . 1 . 10 |
| + | * |
| + | * Primal codes of finite partial functions on positive integers: |
| + | * 1 = { } |
| + | * 2 = 1:1 |
| + | * 3 = 2:1 |
| + | * 4 = 1:2 |
| + | * 5 = 3:1 |
| + | * 6 = 1:1 2:1 |
| + | * 7 = 4:1 |
| + | * 8 = 1:3 |
| + | * 9 = 2:2 |
| + | * 10 = 1:1 3:1 |
| + | * 11 = 5:1 |
| + | * 12 = 1:2 2:1 |
| + | * 13 = 6:1 |
| + | * 14 = 1:1 4:1 |
| + | * 15 = 2:1 3:1 |
| + | * 16 = 1:4 |
| + | * 17 = 7:1 |
| + | * 18 = 1:1 2:2 |
| + | * 19 = 8:1 |
| + | * 20 = 1:2 3:1 |
| + | </pre> |
| + | |
| + | ==A106178== |
| + | |
| + | * [http://oeis.org/wiki/A106178 A106178] |
| + | |
| + | ===Wiki Table=== |
| + | |
| + | {| align="center" style="font-weight:bold; text-align:center; width:90%" |
| + | | || || || || || || || || || || || || || || |
| + | | <font color="red">1</font> |
| + | | |
| + | | <font color="red">1</font> |
| + | |- |
| + | | || || || || || || || || || || || || || |
| + | | <font color="red">2</font> |
| + | | || · || |
| + | | <font color="red">2</font> |
| + | |- |
| + | | || || || || || || || || || || || || |
| + | | <font color="red">3</font> |
| + | | || · || || · || |
| + | | <font color="red">3</font> |
| + | |- |
| + | | || || || || || || || || || || || |
| + | | <font color="red">4</font> |
| + | | || · || || 2 || || · || |
| + | | <font color="red">4</font> |
| + | |- |
| + | | || || || || || || || || || || |
| + | | <font color="red">5</font> |
| + | | || · || || 3 || || 1 || || · || |
| + | | <font color="red">5</font> |
| + | |- |
| + | | || || || || || || || || || |
| + | | <font color="red">6</font> |
| + | | || · || || 1 || || 1 || || 4 || || · || |
| + | | <font color="red">6</font> |
| + | |- |
| + | | || || || || || || || || |
| + | | <font color="red">7</font> |
| + | | || · || || 5 || || 2 || || 9 || || 1 || || · || |
| + | | <font color="red">7</font> |
| + | |- |
| + | | || || || || || || || |
| + | | <font color="red">8</font> |
| + | | || · || || 6 || || 1 || || 1 || || 1 || || 2 || || · || |
| + | | <font color="red">8</font> |
| + | |- |
| + | | || || || || || || |
| + | | <font color="red">9</font> |
| + | | || · || || 7 || || 1 || || 25|| || 1 || || 3 || || 1 || || · || |
| + | | <font color="red">9</font> |
| + | |- |
| + | | || || || || || |
| + | | <font color="red">10</font> |
| + | | || · || || 1 || || 1 || || 36|| || 1 || || 2 || || 1 || || 8 || || · || |
| + | | <font color="red">10</font> |
| + | |- |
| + | | || || || || |
| + | | <font color="red">11</font> |
| + | | || · || || 1 || || 1 || || 49 || || 1 || || 5 || || 1 || || 27 || || 1 || || · || |
| + | | <font color="red">11</font> |
| + | |- |
| + | | || || || |
| + | | <font color="red">12</font> |
| + | | || · || || 10 || || 3 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || · || |
| + | | <font color="red">12</font> |
| + | |- |
| + | | || || |
| + | | <font color="red">13</font> |
| + | | || · || || 11 || || 1 || || 1 || || 2 || || 7 || || 1 || || 125 || || 4 || || 3 || || 1 || || · || |
| + | | <font color="red">13</font> |
| + | |- |
| + | | || |
| + | | <font color="red">14</font> |
| + | | || · || || 3 || || 1 || || 100 || || 1 || || 1 || || 1 || || 216 || || 1 || || 1 || || 1 || || 4 || || · || |
| + | | <font color="red">14</font> |
| + | |- |
| + | | |
| + | | <font color="red">15</font> |
| + | | || · || || 13 || || 2 || || 121 || || 1 || || 3 || || 1 || || 343 || || 1 || || 5 || || 1 || || 9 || || 1 || || · || |
| + | | <font color="red">15</font> |
| + | |- |
| + | | width="3%" | <font color="red">16</font> |
| + | | width="3%" | |
| + | | width="3%" | · |
| + | | width="3%" | |
| + | | width="3%" | 14 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 9 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 10 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 6 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 2 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 2 |
| + | | width="3%" | |
| + | | width="3%" | · |
| + | | width="3%" | |
| + | | width="3%" | <font color="red">16</font> |
| + | |} |
| + | |
| + | ===TeX Smallmatrix=== |
| + | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{smallmatrix} |
| + | &&&&&&&&&&&&&&& {\color{red}1} && {\color{red}1} |
| + | \\ |
| + | &&&&&&&&&&&&&& {\color{red}2} && \cdot & & {\color{red}2} |
| + | \\ |
| + | &&&&&&&&&&&&& {\color{red}3} && \cdot && \cdot && {\color{red}3} |
| + | \\ |
| + | &&&&&&&&&&&& {\color{red}4} && \cdot && 2 && \cdot && {\color{red}4} |
| + | \\ |
| + | &&&&&&&&&&& {\color{red}5} && \cdot && 3 && 1 && \cdot && {\color{red}5} |
| + | \\ |
| + | &&&&&&&&&& {\color{red}6} && \cdot && 1 && 1 && 4 && \cdot && {\color{red}6} |
| + | \\ |
| + | &&&&&&&&& {\color{red}7} && \cdot && 5 && 2 && 9 && 1 && \cdot && {\color{red}7} |
| + | \\ |
| + | &&&&&&&& {\color{red}8} && \cdot && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}8} |
| + | \\ |
| + | &&&&&&& {\color{red}9} && \cdot && 7 && 1 && 25 && 1 && 3 && 1 && \cdot && {\color{red}9} |
| + | \\ |
| + | &&&&&& {\color{red}10} && \cdot && 1 && 1 && 36 && 1 && 2 && 1 && 8 && \cdot && {\color{red}10} |
| + | \\ |
| + | &&&&& {\color{red}11} && \cdot && 1 && 1 && 49 && 1 && 5 && 1 && 27 && 1 && \cdot && {\color{red}11} |
| + | \\ |
| + | &&&& {\color{red}12} && \cdot && 10 && 3 && 1 && 1 && 6 && 1 && 1 && 1 && 2 && \cdot && {\color{red}12} |
| + | \\ |
| + | &&& {\color{red}13} && \cdot && 11 && 1 && 1 && 2 && 7 && 1 && 125 && 4 && 3 && 1 && \cdot && {\color{red}13} |
| + | \\ |
| + | && {\color{red}14} && \cdot && 3 && 1 && 100 && 1 && 1 && 1 && 216 && 1 && 1 && 1 && 4 && \cdot && {\color{red}14} |
| + | \\ |
| + | & {\color{red}15} && \cdot && 13 && 2 && 121 && 1 && 3 && 1 && 343 && 1 && 5 && 1 && 9 && 1 && \cdot && {\color{red}15} |
| + | \\ |
| + | {\color{red}16} && \cdot && 14 && 1 && 9 && 1 && 10 && 1 && 1 && 1 && 6 && 1 && 2 && 1 && 2 && \cdot && {\color{red}16} |
| + | \end{smallmatrix}</math> |
| + | |} |
| + | |
| + | ===ASCII=== |
| + | |
| + | <pre> |
| + | Example |
| + | |
| + | * n o m |
| + | * \ / |
| + | * 1 . 1 |
| + | * \ / \ / |
| + | * 2 . . 2 |
| + | * \ / \ / \ / |
| + | * 3 . . . 3 |
| + | * \ / \ / \ / \ / |
| + | * 4 . . 2 . . 4 |
| + | * \ / \ / \ / \ / \ / |
| + | * 5 . . 3 . 1 . . 5 |
| + | * \ / \ / \ / \ / \ / \ / |
| + | * 6 . . 1 . 1 . 4 . . 6 |
| + | * \ / \ / \ / \ / \ / \ / \ / |
| + | * 7 . . 5 . 2 . 9 . 1 . . 7 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 8 . . 6 . 1 . 1 . 1 . 2 . . 8 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 9 . . 7 . 1 . 25. 1 . 3 . 1 . . 9 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 10 . . 1 . 1 . 36. 1 . 2 . 1 . 8 . . 10 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 11 . . 1 . 1 . 49. 1 . 5 . 1 . 27. 1 . . 11 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 12 . . 10. 3 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . . 12 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 13 . . 11. 1 . 1 . 2 . 7 . 1 .125. 4 . 3 . 1 . . 13 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 14 . . 3 . 1 .100. 1 . 1 . 1 .216. 1 . 1 . 1 . 4 . . 14 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 15 . . 13. 2 .121. 1 . 3 . 1 .343. 1 . 5 . 1 . 9 . 1 . . 15 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 16 . . 14. 1 . 9 . 1 . 10. 1 . 1 . 1 . 6 . 1 . 2 . 1 . 2 . . 16 |
| + | </pre> |
| + | |
| + | ==A108352== |
| + | |
| + | * [http://oeis.org/wiki/A108352 A108352] |
| + | |
| + | ===Links=== |
| + | |
| + | * Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002846.html Primal Code Characteristic, n = 1 to 1000] |
| + | * Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002847.html Primal Code Characteristic, n = 1001 to 2000] |
| + | * Jon Awbrey, [http://stderr.org/pipermail/inquiry/2005-July/002853.html Primal Code Characteristic, n = 2001 to 3000] |
| + | |
| + | ===TeX Array=== |
| + | |
| + | {| align="center" |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | a(1) |
| + | & = & 1 |
| + | & \text{because} & (\circ~ 1)^1 |
| + | & = & (\circ~ \varnothing)^1 |
| + | & = & 1. |
| + | \\ |
| + | a(2) |
| + | & = & 0 |
| + | & \text{because} & (\circ~ 2)^k |
| + | & = & (\circ~ 1\!:\!1)^k |
| + | & = & 2, |
| + | & \text{for all}~ k > 0. |
| + | \\ |
| + | a(3) |
| + | & = & 2 |
| + | & \text{because} & (\circ~ 3)^2 |
| + | & = & (\circ~ 2\!:\!1)^2 |
| + | & = & 1. |
| + | \\ |
| + | a(4) |
| + | & = & 2 |
| + | & \text{because} & (\circ~ 4 )^2 |
| + | & = & (\circ~ 1\!:\!2)^2 |
| + | & = &1. |
| + | \\ |
| + | a(5) |
| + | & = & 2 |
| + | & \text{because} & (\circ~ 5)^2 |
| + | & = & (\circ~ 3\!:\!1)^2 |
| + | & = & 1. |
| + | \\ |
| + | a(6) |
| + | & = & 0 |
| + | & \text{because} & (\circ~ 6)^k |
| + | & = & (\circ~ 1\!:\!1 ~~ 2\!:\!1)^k |
| + | & = & 6, |
| + | & \text{for all}~ k > 0. |
| + | \\ |
| + | a(7) |
| + | & = & 2 |
| + | & \text{because} & (\circ~ 7)^2 |
| + | & = & (\circ~ 4\!:\!1)^1 |
| + | & = & 1. |
| + | \\ |
| + | a(8) |
| + | & = & 2 |
| + | & \text{because} & (\circ~ 8)^2 |
| + | & = & (\circ~ 1\!:\!3)^1 |
| + | & = & 1. |
| + | \\ |
| + | a(9) |
| + | & = & 0 |
| + | & \text{because} & (\circ~ 9)^k |
| + | & = & (\circ~ 2\!:\!2)^k |
| + | & = & 9, |
| + | & \text{for all}~ k > 0. |
| + | \\ |
| + | a(10) |
| + | & = & 0 |
| + | & \text{because} & (\circ~ 10)^k |
| + | & = & (\circ~ 1\!:\!1 ~~ 3\!:\!1)^k |
| + | & = & 10, |
| + | & \text{for all}~ k > 0. |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | ===ASCII=== |
| + | |
| + | <pre> |
| + | Example |
| + | |
| + | * a(1) = 1 because (1 o)^1 = ({ } o)^1 = 1. |
| + | * a(2) = 0 because (2 o)^k = (1:1 o)^k = 2, for all positive k. |
| + | * a(3) = 2 because (3 o)^2 = (2:1 o)^2 = 1. |
| + | * a(4) = 2 because (4 o)^2 = (1:2 o)^2 = 1. |
| + | * a(5) = 2 because (5 o)^2 = (3:1 o)^2 = 1. |
| + | * a(6) = 0 because (6 o)^k = (1:1 2:1 o)^k = 6, for all positive k. |
| + | * a(7) = 2 because (7 o)^2 = (4:1 o)^1 = 1. |
| + | * a(8) = 2 because (8 o)^2 = (1:3 o)^1 = 1. |
| + | * a(9) = 0 because (9 o)^k = (2:2 o)^k = 9, for all positive k. |
| + | * a(10) = 0 because (10 o)^k = (1:1 3:1 o)^k = 10, for all positive k. |
| + | * Detail of calculation for compositional powers of 12: |
| + | * (12 o)^2 = (1:2 2:1) o (1:2 2:1) = (1:1 2:2) = 18 |
| + | * (12 o)^3 = (1:1 2:2) o (1:2 2:1) = (1:2 2:1) = 12 |
| + | * Detail of calculation for compositional powers of 20: |
| + | * (20 o)^2 = (1:2 3:1) o (1:2 3:1) = (3:2) = 25 |
| + | * (20 o)^3 = (3:2) o (1:2 3:1) = 1 |
| + | </pre> |
| + | |
| + | ==A108371== |
| + | |
| + | * [http://oeis.org/wiki/A108371 A108371] |
| + | |
| + | ===Wiki Table=== |
| + | |
| + | {| align="center" style="font-weight:bold; text-align:center; width:90%" |
| + | | || || || || || || || || || || || || || || |
| + | | <font color="red">1</font> |
| + | | |
| + | | <font color="red">1</font> |
| + | |- |
| + | | || || || || || || || || || || || || || |
| + | | <font color="red">2</font> |
| + | | || 1 || |
| + | | <font color="red">2</font> |
| + | |- |
| + | | || || || || || || || || || || || || |
| + | | <font color="red">3</font> |
| + | | || 2 || || 1 || |
| + | | <font color="red">3</font> |
| + | |- |
| + | | || || || || || || || || || || || |
| + | | <font color="red">4</font> |
| + | | || 3 || || 2 || || 1 || |
| + | | <font color="red">4</font> |
| + | |- |
| + | | || || || || || || || || || || |
| + | | <font color="red">5</font> |
| + | | || 4 || || 1 || || 2 || || 1 || |
| + | | <font color="red">5</font> |
| + | |- |
| + | | || || || || || || || || || |
| + | | <font color="red">6</font> |
| + | | || 5 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">6</font> |
| + | |- |
| + | | || || || || || || || || |
| + | | <font color="red">7</font> |
| + | | || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">7</font> |
| + | |- |
| + | | || || || || || || || |
| + | | <font color="red">8</font> |
| + | | || 7 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">8</font> |
| + | |- |
| + | | || || || || || || |
| + | | <font color="red">9</font> |
| + | | || 8 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">9</font> |
| + | |- |
| + | | || || || || || |
| + | | <font color="red">10</font> |
| + | | || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">10</font> |
| + | |- |
| + | | || || || || |
| + | | <font color="red">11</font> |
| + | | || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">11</font> |
| + | |- |
| + | | || || || |
| + | | <font color="red">12</font> |
| + | | || 11|| || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">12</font> |
| + | |- |
| + | | || || |
| + | | <font color="red">13</font> |
| + | | || 12|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">13</font> |
| + | |- |
| + | | || |
| + | | <font color="red">14</font> |
| + | | || 13|| || 18|| || 1 || || 10|| || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">14</font> |
| + | |- |
| + | | |
| + | | <font color="red">15</font> |
| + | | || 14 || || 1 || || 12 || || 1 || || 10 || || 9 || || 1 || || 1 || || 6 || || 1 || || 1 || || 1 || || 2 || || 1 || |
| + | | <font color="red">15</font> |
| + | |- |
| + | | width="3%" | <font color="red">16</font> |
| + | | width="3%" | |
| + | | width="3%" | 15 |
| + | | width="3%" | |
| + | | width="3%" | 14 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 18 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 10 |
| + | | width="3%" | |
| + | | width="3%" | 9 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 6 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | 2 |
| + | | width="3%" | |
| + | | width="3%" | 1 |
| + | | width="3%" | |
| + | | width="3%" | <font color="red">16</font> |
| + | |} |
| + | |
| + | ===ASCII=== |
| + | |
| + | <pre> |
| + | Example |
| + | |
| + | * Table: T(n,k) = (n o)^k |
| + | * T(n,k) |
| + | * \ / |
| + | * 1 . 1 |
| + | * \ / \ / |
| + | * 2 . 1 . 2 |
| + | * \ / \ / \ / |
| + | * 3 . 2 . 1 . 3 |
| + | * \ / \ / \ / \ / |
| + | * 4 . 3 . 2 . 1 . 4 |
| + | * \ / \ / \ / \ / \ / |
| + | * 5 . 4 . 1 . 2 . 1 . 5 |
| + | * \ / \ / \ / \ / \ / \ / |
| + | * 6 . 5 . 1 . 1 . 2 . 1 . 6 |
| + | * \ / \ / \ / \ / \ / \ / \ / |
| + | * 7 . 6 . 1 . 1 . 1 . 2 . 1 . 7 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 8 . 7 . 6 . 1 . 1 . 1 . 2 . 1 . 8 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 9 . 8 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 9 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 10 . 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 10 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 11 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 11 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 12 . 11. 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 12 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 13 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 13 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 14 . 13. 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 14 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 15 . 14. 1 . 12. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 15 |
| + | * \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / |
| + | * 16 . 15. 14. 1 . 18. 1 . 10. 9 . 1 . 1 . 6 . 1 . 1 . 1 . 2 . 1 . 16 |
| + | </pre> |
| + | |
| + | ==A109300== |
| + | |
| + | * [http://oeis.org/wiki/A109300 A109300] |
| + | |
| + | ===JPEG=== |
| + | |
| + | {| align="center" border="1" cellpadding="10" |
| + | | |
| + | <p>[[Image:Rote 3 Big.jpg|40px]]</p><br> |
| + | <p><math>\begin{array}{l} 2\!:\!1 \\ 3 \end{array}</math></p> |
| + | | |
| + | <p>[[Image:Rote 4 Big.jpg|65px]]</p><br> |
| + | <p><math>\begin{array}{l} 1\!:\!2 \\ 4 \end{array}</math></p> |
| + | | |
| + | <p>[[Image:Rote 6 Big.jpg|80px]]</p><br> |
| + | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}</math></p> |
| + | | |
| + | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> |
| + | <p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p> |
| + | | |
| + | <p>[[Image:Rote 12 Big.jpg|105px]]</p><br> |
| + | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}</math></p> |
| + | | |
| + | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> |
| + | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p> |
| + | | |
| + | <p>[[Image:Rote 36 Big.jpg|145px]]</p><br> |
| + | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p> |
| + | |} |
| + | |
| + | ===ASCII=== |
| + | |
| + | <pre> |
| + | Example |
| + | |
| + | * Table of Rotes and Primal Functions for Positive Integers of Rote Height 2 |
| + | * |
| + | * o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o o-o |
| + | * | | | | | | | | | | | | |
| + | * o-o o-o o-o o-o o---o o-o o-o o-o o---o o-o o---o |
| + | * | | | | | | | | | | | |
| + | * O O O===O O O=====O O===O O=====O |
| + | * |
| + | * 2:1 1:2 1:1 2:1 2:2 1:2 2:1 1:1 2:2 1:2 2:2 |
| + | * |
| + | * 3 4 6 9 12 18 36 |
| + | * |
| + | </pre> |
| + | |
| + | ==A109301== |
| + | |
| + | * [http://oeis.org/wiki/A109301 A109301] |
| + | |
| + | ===Example=== |
| + | |
| + | : <math>802701 = 9 \cdot 89189 = \text{p}_2^2 \text{p}_{8638}^1</math> |
| + | |
| + | : <math>\text{Writing}~ (\operatorname{prime}(i))^j ~\text{as}~ i\!:\!j, ~\text{we have:}</math> |
| + | |
| + | : <math>\begin{array}{lllll} |
| + | 802701 |
| + | & = & 9 \cdot 89189 |
| + | & = & 2\!:\!2 ~~ 8638\!:\!1 |
| + | \\ |
| + | 8638 |
| + | & = & 2 \cdot 7 \cdot 617 |
| + | & = & 1\!:\!1 ~~ 4\!:\!1 ~~ 113\!:\!1 |
| + | \\ |
| + | 113 |
| + | & & |
| + | & = & 30\!:\!1 |
| + | \\ |
| + | 30 |
| + | & = & 2 \cdot 3 \cdot 5 |
| + | & = & 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 |
| + | \\ |
| + | 4 |
| + | & & |
| + | & = & 1\!:\!2 |
| + | \\ |
| + | 3 |
| + | & & |
| + | & = & 2\!:\!1 |
| + | \\ |
| + | 2 |
| + | & & |
| + | & = & 1\!:\!1 |
| + | \end{array}</math> |
| + | |
| + | : <math>\text{So the rote of 802701 is the following graph:}\!</math> |
| + | |
| + | :{| border="1" cellpadding="20" |
| + | | [[Image:Rote 802701 Big.jpg|330px]] |
| + | |} |
| + | |
| + | : <math>\text{By inspection, the rote height of 802701 is 6.}\!</math> |
| + | |
| + | ===JPEG=== |
| + | |
| + | {| align="center" border="1" cellpadding="6" |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 1 Big.jpg|20px]]</p><br> |
| + | <p><math>1\!</math></p><br> |
| + | <p><math>a(1) ~=~ 0</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 2 Big.jpg|40px]]</p><br> |
| + | <p><math>\text{p}\!</math></p><br> |
| + | <p><math>a(2) ~=~ 1</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 3 Big.jpg|40px]]</p><br> |
| + | <p><math>\text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(3) ~=~ 2</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 4 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(4) ~=~ 2</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 5 Big.jpg|40px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(5) ~=~ 3</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 6 Big.jpg|80px]]</p><br> |
| + | <p><math>\text{p} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(6) ~=~ 2</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 7 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(7) ~=~ 3</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 8 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(8) ~=~ 3</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> |
| + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(9) ~=~ 2</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 10 Big.jpg|80px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(10) ~=~ 3</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 11 Big.jpg|40px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(11) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 12 Big.jpg|105px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(12) ~=~ 2</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 13 Big.jpg|80px]]</p><br> |
| + | <p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(13) ~=~ 3</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 14 Big.jpg|105px]]</p><br> |
| + | <p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(14) ~=~ 3</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 15 Big.jpg|80px]]</p><br> |
| + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(15) ~=~ 3</math></p> |
| + | |- |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 16 Big.jpg|90px]]</p><br> |
| + | <p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(16) ~=~ 3</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 17 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(17) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> |
| + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(18) ~=~ 2</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 19 Big.jpg|65px]]</p><br> |
| + | <p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(19) ~=~ 4</math></p> |
| + | | valign="bottom" | |
| + | <p>[[Image:Rote 20 Big.jpg|105px]]</p><br> |
| + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(20) ~=~ 3</math></p> |
| + | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 21 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 21 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}</math></p> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(21) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 22 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 22 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(22) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 23 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 23 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 9\!:\!1 \\ 23 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(23) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 24 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 24 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}</math></p> | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(24) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 25 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 25 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 3\!:\!2 \\ 25 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(25) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 26 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 26 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(26) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 27 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 27 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!3 \\ 27 \end{array}</math></p> | + | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(27) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 28 Big.jpg|130px]]</p><br> | | <p>[[Image:Rote 28 Big.jpg|130px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}</math></p> | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(28) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 29 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 29 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 10\!:\!1 \\ 29 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(29) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 30 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 30 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(30) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 31 Big.jpg|40px]]</p><br> | | <p>[[Image:Rote 31 Big.jpg|40px]]</p><br> |
− | <p><math>\begin{array}{l} 11\!:\!1 \\ 31 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br> |
| + | <p><math>a(31) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 32 Big.jpg|65px]]</p><br> | | <p>[[Image:Rote 32 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!5 \\ 32 \end{array}</math></p> | + | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(32) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 33 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 33 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}</math></p> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(33) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 34 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 34 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(34) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 35 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 35 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(35) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 36 Big.jpg|145px]]</p><br> | | <p>[[Image:Rote 36 Big.jpg|145px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}</math></p> | + | <p><math>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(36) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 37 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 37 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 12\!:\!1 \\ 37 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(37) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 38 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 38 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(38) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 39 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 39 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}</math></p> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(39) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 40 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 40 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}</math></p> | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(40) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 41 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 41 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 13\!:\!1 \\ 41 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(41) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 42 Big.jpg|145px]]</p><br> | | <p>[[Image:Rote 42 Big.jpg|145px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(42) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 43 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 43 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 14\!:\!1 \\ 43 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(43) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 44 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 44 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}</math></p> | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(44) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 45 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 45 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}</math></p> | + | <p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(45) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 46 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 46 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(46) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 47 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 47 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 15\!:\!1 \\ 47 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(47) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 48 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 48 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}</math></p> | + | <p><math>\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!</math></p><br> |
| + | <p><math>a(48) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 49 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 49 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 4\!:\!2 \\ 49 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(49) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 50 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 50 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
| + | <p><math>a(50) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 51 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 51 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}</math></p> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(51) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 52 Big.jpg|145px]]</p><br> | | <p>[[Image:Rote 52 Big.jpg|145px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}</math></p> | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(52) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 53 Big.jpg|90px]]</p><br> | | <p>[[Image:Rote 53 Big.jpg|90px]]</p><br> |
− | <p><math>\begin{array}{l} 16\!:\!1 \\ 53 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> |
| + | <p><math>a(53) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 54 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 54 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(54) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 55 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 55 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(55) ~=~ 4</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 56 Big.jpg|130px]]</p><br> | | <p>[[Image:Rote 56 Big.jpg|130px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}</math></p> | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(56) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 57 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 57 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}</math></p> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(57) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 58 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 58 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(58) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 59 Big.jpg|65px]]</p><br> | | <p>[[Image:Rote 59 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 17\!:\!1 \\ 59 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br> |
| + | <p><math>a(59) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 60 Big.jpg|155px]]</p><br> | | <p>[[Image:Rote 60 Big.jpg|155px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}</math></p> | + | <p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(60) ~=~ 3</math></p> |
| |} | | |} |
| | | |
Line 1,673: |
Line 2,951: |
| * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 = | | * while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 = |
| * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly | | * (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly |
− | * 7.840858554516122655953405327738 x 10^371. | + | * 7.840858554516122655953405327738 x 10^371. |
| + | |
| + | Example |
| + | |
| + | * Writing (prime(i))^j as i:j, we have: |
| + | * 802701 = 2:2 8638:1 |
| + | * 8638 = 1:1 4:1 113:1 |
| + | * 113 = 30:1 |
| + | * 30 = 1:1 2:1 3:1 |
| + | * 4 = 1:2 |
| + | * 3 = 2:1 |
| + | * 2 = 1:1 |
| + | * 1 = { } |
| + | * So rote(802701) is the graph: |
| + | * |
| + | * o-o |
| + | * | |
| + | * o-o o-o |
| + | * | | |
| + | * o-o o-o o-o o-o |
| + | * | | | | |
| + | * o-o o===o===o-o |
| + | * | | |
| + | * o-o o-o o-o o-o o---------o |
| + | * | | | | | |
| + | * o---o o===o=====o---------o |
| + | * | | |
| + | * O=======O |
| + | * |
| + | * Therefore rhig(802701) = 6. |
| </pre> | | </pre> |
| | | |