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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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| | [[Image:Riff 23 Big.jpg|65px]] | | | [[Image:Riff 23 Big.jpg|65px]] |
| | [[Image:Rote 23 Big.jpg|80px]] | | | [[Image:Rote 23 Big.jpg|80px]] |
− | |}
| |
− | |}
| |
− |
| |
− | ===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="10"
| |
− | |+ 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" | | + | | <math>25\!</math> |
− | <p>[[Image:Riff 6 Big.jpg|65px]]</p><br> | + | | |
− | <p><math>\text{p} \text{p}_{\text{p}}\!</math></p><br> | + | <math>\begin{array}{lll} |
− | <p><math>a(6) ~=~ 3</math></p>
| + | \text{p}_3^2 |
− | | valign="bottom" | | + | & = & \text{p}_{\text{p}_2^1}^{\text{p}_1^1} |
− | <p>[[Image:Riff 7 Big.jpg|65px]]</p><br>
| + | \\[12pt] |
− | <p><math>\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(7) ~=~ 3</math></p>
| + | \end{array}</math> |
− | | valign="bottom" | | + | | <math>\text{p}_{\text{p}_{\text{p}}}^{\text{p}}\!</math> |
− | <p>[[Image:Riff 8 Big.jpg|65px]]</p><br>
| + | | [[Image:Riff 25 Big.jpg|65px]] |
− | <p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br> | + | | [[Image:Rote 25 Big.jpg|80px]] |
− | <p><math>a(8) ~=~ 3</math></p>
| + | |- |
− | | valign="bottom" | | + | | <math>27\!</math> |
− | <p>[[Image:Riff 9 Big.jpg|40px]]</p><br>
| + | | |
− | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> | + | <math>\begin{array}{lll} |
− | <p><math>a(9) ~=~ 3</math></p>
| + | \text{p}_2^3 |
− | | valign="bottom" | | + | & = & \text{p}_{\text{p}_1^1}^{\text{p}_2^1} |
− | <p>[[Image:Riff 10 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(10) ~=~ 4</math></p>
| + | \end{array}</math> |
| + | | <math>\text{p}_{\text{p}}^{\text{p}_{\text{p}}}\!</math> |
| + | | [[Image:Riff 27 Big.jpg|65px]] |
| + | | [[Image:Rote 27 Big.jpg|80px]] |
| + | |- |
| + | | <math>32\!</math> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \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> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \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> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \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> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \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> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \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> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \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> |
| + | | |
| + | <math>\begin{array}{lll} |
| + | \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]] |
| + | |} |
| + | |} |
| + | |
| + | ===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="10" |
| + | |+ style="height:25px" | <math>a(n) = \text{Number of Nodes in the Riff of}~ n</math> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 11 Big.jpg|90px]]</p><br> | + | <p> </p><br> |
− | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br> | + | <p><math>1\!</math></p><br> |
− | <p><math>a(11) ~=~ 4</math></p> | + | <p><math>a(1) ~=~ 0</math></p> |
− | | valign="bottom" | | + | | valign="bottom" | |
− | <p>[[Image:Riff 12 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> |
− | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> | + | <p><math>\text{p}\!</math></p><br> |
− | <p><math>a(12) ~=~ 4</math></p> | + | <p><math>a(2) ~=~ 1</math></p> |
− | | valign="bottom" | | + | | valign="bottom" | |
− | <p>[[Image:Riff 13 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> |
− | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_\text{p}\!</math></p><br> |
− | <p><math>a(13) ~=~ 4</math></p> | + | <p><math>a(3) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 14 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 4 Big.jpg|40px]]</p><br> |
− | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br> | + | <p><math>\text{p}^\text{p}\!</math></p><br> |
− | <p><math>a(14) ~=~ 4</math></p> | + | <p><math>a(4) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | <p><math>a(15) ~=~ 5</math></p> | + | <p><math>a(5) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 16 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 6 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br> | + | <p><math>\text{p} \text{p}_{\text{p}}\!</math></p><br> |
− | <p><math>a(16) ~=~ 3</math></p> | + | <p><math>a(6) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 17 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 7 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}^{\text{p}}}\!</math></p><br> |
− | <p><math>a(17) ~=~ 4</math></p> | + | <p><math>a(7) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 18 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 8 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> | + | <p><math>\text{p}^{\text{p}_{\text{p}}}\!</math></p><br> |
− | <p><math>a(18) ~=~ 4</math></p> | + | <p><math>a(8) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 19 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 9 Big.jpg|40px]]</p><br> |
− | <p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br> | + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> |
− | <p><math>a(19) ~=~ 4</math></p> | + | <p><math>a(9) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 20 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | <p><math>a(20) ~=~ 5</math></p> | + | <p><math>a(10) ~=~ 4</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 21 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 11 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}_{\text{p}_{\text{p}}}}\!</math></p><br> |
− | <p><math>a(21) ~=~ 5</math></p> | + | <p><math>a(11) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 22 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 12 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p}^\text{p} \text{p}_\text{p}\!</math></p><br> |
− | <p><math>a(22) ~=~ 5</math></p> | + | <p><math>a(12) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 23 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 13 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}}}\!</math></p><br> |
− | <p><math>a(23) ~=~ 4</math></p> | + | <p><math>a(13) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 24 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 14 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> | + | <p><math>\text{p} \text{p}_{\text{p}^{\text{p}}}\!</math></p><br> |
− | <p><math>a(24) ~=~ 5</math></p> | + | <p><math>a(14) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 25 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | <p><math>a(25) ~=~ 4</math></p> | + | <p><math>a(15) ~=~ 5</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 26 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 16 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | + | <p><math>\text{p}^{\text{p}^{\text{p}}}\!</math></p><br> |
− | <p><math>a(26) ~=~ 5</math></p> | + | <p><math>a(16) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 27 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 17 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}_{\text{p}^{\text{p}}}}\!</math></p><br> |
− | <p><math>a(27) ~=~ 4</math></p> | + | <p><math>a(17) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 28 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 18 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
− | <p><math>a(28) ~=~ 5</math></p> | + | <p><math>a(18) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 29 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 19 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}^{\text{p}_{\text{p}}}}\!</math></p><br> |
− | <p><math>a(29) ~=~ 5</math></p> | + | <p><math>a(19) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 20 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | <p><math>a(30) ~=~ 6</math></p> | + | <p><math>a(20) ~=~ 5</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 31 Big.jpg|115px]]</p><br> | + | <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>\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
− | <p><math>a(31) ~=~ 5</math></p> | + | <p><math>a(21) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 32 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 22 Big.jpg|115px]]</p><br> |
− | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(32) ~=~ 4</math></p> | + | <p><math>a(22) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 33 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 23 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
− | <p><math>a(33) ~=~ 6</math></p> | + | <p><math>a(23) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 34 Big.jpg|115px]]</p><br> | + | <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>\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!</math></p><br> |
− | <p><math>a(34) ~=~ 5</math></p> | + | <p><math>a(24) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 35 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 25 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
− | <p><math>a(35) ~=~ 6</math></p> | + | <p><math>a(25) ~=~ 4</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 36 Big.jpg|65px]]</p><br> | + | <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>\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(36) ~=~ 5</math></p> | + | <p><math>a(26) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 37 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 27 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> | + | <p><math>\text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(37) ~=~ 5</math></p> | + | <p><math>a(27) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 38 Big.jpg|115px]]</p><br> | + | <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>\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
− | <p><math>a(38) ~=~ 5</math></p> | + | <p><math>a(28) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 39 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 29 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(39) ~=~ 6</math></p> | + | <p><math>a(29) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 40 Big.jpg|135px]]</p><br> | + | <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>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(40) ~=~ 6</math></p> | + | <p><math>a(30) ~=~ 6</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 41 Big.jpg|90px]]</p><br> | + | <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>\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!</math></p><br> |
− | <p><math>a(41) ~=~ 5</math></p> | + | <p><math>a(31) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 42 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 32 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | + | <p><math>\text{p}^{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(42) ~=~ 6</math></p> | + | <p><math>a(32) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 43 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 33 Big.jpg|115px]]</p><br> |
− | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(43) ~=~ 5</math></p> | + | <p><math>a(33) ~=~ 6</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 44 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 34 Big.jpg|115px]]</p><br> |
− | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
− | <p><math>a(44) ~=~ 6</math></p> | + | <p><math>a(34) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 45 Big.jpg|90px]]</p><br> | + | <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>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
− | <p><math>a(45) ~=~ 6</math></p> | + | <p><math>a(35) ~=~ 6</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 46 Big.jpg|90px]]</p><br> | + | <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>\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
− | <p><math>a(46) ~=~ 5</math></p> | + | <p><math>a(36) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 47 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 37 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(47) ~=~ 6</math></p> | + | <p><math>a(37) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 48 Big.jpg|65px]]</p><br> | + | <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>\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(48) ~=~ 5</math></p> | + | <p><math>a(38) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 49 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 39 Big.jpg|115px]]</p><br> |
− | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(49) ~=~ 4</math></p> | + | <p><math>a(39) ~=~ 6</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 50 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 40 Big.jpg|135px]]</p><br> |
− | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(50) ~=~ 5</math></p> | + | <p><math>a(40) ~=~ 6</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 51 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 41 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(51) ~=~ 6</math></p> | + | <p><math>a(41) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 52 Big.jpg|90px]]</p><br> | + | <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>\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
− | <p><math>a(52) ~=~ 6</math></p> | + | <p><math>a(42) ~=~ 6</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 53 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 43 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
− | <p><math>a(53) ~=~ 4</math></p> | + | <p><math>a(43) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 54 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 44 Big.jpg|115px]]</p><br> |
− | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(54) ~=~ 5</math></p> | + | <p><math>a(44) ~=~ 6</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 45 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(55) ~=~ 7</math></p> | + | <p><math>a(45) ~=~ 6</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 56 Big.jpg|135px]]</p><br> | + | <p>[[Image:Riff 46 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!</math></p><br> |
− | <p><math>a(56) ~=~ 6</math></p> | + | <p><math>a(46) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 57 Big.jpg|115px]]</p><br> | + | <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>\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(57) ~=~ 6</math></p> | + | <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" | | | | valign="bottom" | |
− | <p>[[Image:Riff 58 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 49 Big.jpg|65px]]</p><br> |
− | <p><math>\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}^\text{p}}^\text{p}\!</math></p><br> |
− | <p><math>a(58) ~=~ 6</math></p> | + | <p><math>a(49) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 59 Big.jpg|115px]]</p><br> | + | <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>\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!</math></p><br> |
− | <p><math>a(59) ~=~ 5</math></p> | + | <p><math>a(50) ~=~ 5</math></p> |
| + | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 60 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 51 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>\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!</math></p><br> |
− | <p><math>a(60) ~=~ 7</math></p> | + | <p><math>a(51) ~=~ 6</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" | | | | valign="bottom" | |
− | <p> </p><br> | + | <p>[[Image:Riff 52 Big.jpg|90px]]</p><br> |
− | <p><math>1\!</math></p><br> | + | <p><math>\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(0) ~=~ 1</math></p> | + | <p><math>a(52) ~=~ 6</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> | + | <p>[[Image:Riff 53 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}^{\text{p}^\text{p}}}\!</math></p><br> |
− | <p><math>a(1) ~=~ 2</math></p> | + | <p><math>a(53) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> | + | <p>[[Image:Riff 54 Big.jpg|90px]]</p><br> |
− | <p><math>\text{p}_\text{p}\!</math></p><br> | + | <p><math>\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(2) ~=~ 3</math></p> | + | <p><math>a(54) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> | + | <p>[[Image:Riff 55 Big.jpg|115px]]</p><br> |
− | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | + | <p><math>\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
− | <p><math>a(3) ~=~ 5</math></p> | + | <p><math>a(55) ~=~ 7</math></p> |
| + | |- |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> | + | <p>[[Image:Riff 56 Big.jpg|135px]]</p><br> |
− | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> | + | <p><math>\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
− | <p><math>a(4) ~=~ 10</math></p> | + | <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" | | | | valign="bottom" | |
− | <p>[[Image:Riff 15 Big.jpg|90px]]</p><br> | + | <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>\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!</math></p><br> |
− | <p><math>a(5) ~=~ 15</math></p> | + | <p><math>a(59) ~=~ 5</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
− | <p>[[Image:Riff 30 Big.jpg|115px]]</p><br> | + | <p>[[Image:Riff 60 Big.jpg|115px]]</p><br> |
− | <p><math>\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> | + | <p><math>\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
− | <p><math>a(6) ~=~ 30</math></p> | + | <p><math>a(60) ~=~ 7</math></p> |
| |} | | |} |
| | | |
− | ==A106177== | + | ==A062860== |
| | | |
− | * [http://oeis.org/wiki/A106177 A106177] | + | * [http://oeis.org/wiki/A062860 A062860] |
| | | |
− | ===Primal Codes of Finite Partial Functions on Positive Integers=== | + | ===Wiki + TeX + JPEG=== |
| | | |
− | {| align="center" | + | {| 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> |
− | <math>\begin{array}{rcl} | + | | valign="bottom" | |
− | 1 & = & \varnothing \\
| + | <p> </p><br> |
− | 2 & = & 1\!:\!1 \\
| + | <p><math>1\!</math></p><br> |
− | 3 & = & 2\!:\!1 \\
| + | <p><math>a(0) ~=~ 1</math></p> |
− | 4 & = & 1\!:\!2 \\
| + | | valign="bottom" | |
− | 5 & = & 3\!:\!1 \\
| + | <p>[[Image:Riff 2 Big.jpg|20px]]</p><br> |
− | 6 & = & 1\!:\!1 ~~ 2\!:\!1 \\
| + | <p><math>\text{p}\!</math></p><br> |
− | 7 & = & 4\!:\!1 \\
| + | <p><math>a(1) ~=~ 2</math></p> |
− | 8 & = & 1\!:\!3 \\
| + | | valign="bottom" | |
− | 9 & = & 2\!:\!2 \\
| + | <p>[[Image:Riff 3 Big.jpg|40px]]</p><br> |
− | 10 & = & 1\!:\!1 ~~ 3\!:\!1 \\
| + | <p><math>\text{p}_\text{p}\!</math></p><br> |
− | 11 & = & 5\!:\!1 \\
| + | <p><math>a(2) ~=~ 3</math></p> |
− | 12 & = & 1\!:\!2 ~~ 2\!:\!1 \\
| + | | valign="bottom" | |
− | 13 & = & 6\!:\!1 \\
| + | <p>[[Image:Riff 5 Big.jpg|65px]]</p><br> |
− | 14 & = & 1\!:\!1 ~~ 4\!:\!1 \\
| + | <p><math>\text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | 15 & = & 2\!:\!1 ~~ 3\!:\!1 \\
| + | <p><math>a(3) ~=~ 5</math></p> |
− | 16 & = & 1\!:\!4 \\
| + | | valign="bottom" | |
− | 17 & = & 7\!:\!1 \\
| + | <p>[[Image:Riff 10 Big.jpg|90px]]</p><br> |
− | 18 & = & 1\!:\!1 ~~ 2\!:\!2 \\
| + | <p><math>\text{p} \text{p}_{\text{p}_{\text{p}}}\!</math></p><br> |
− | 19 & = & 8\!:\!1 \\
| + | <p><math>a(4) ~=~ 10</math></p> |
− | 20 & = & 1\!:\!2 ~~ 3\!:\!1
| + | |- |
− | \end{array}</math>
| + | | 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> |
| |} | | |} |
| | | |
− | ===Wiki Table=== | + | ==A106177== |
| + | |
| + | * [http://oeis.org/wiki/A106177 A106177] |
| + | |
| + | ===Primal Codes of Finite Partial Functions on Positive Integers=== |
| | | |
− | {| align="center" style="font-weight:bold; text-align:center" | + | {| align="center" |
− | | || || || || || || || ||
| |
− | | <font color="red">1</font>
| |
| | | | | |
− | | <font color="red">1</font> | + | <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> | | | <font color="red">2</font> |
Line 2,350: |
Line 2,505: |
| * [http://oeis.org/wiki/A109301 A109301] | | * [http://oeis.org/wiki/A109301 A109301] |
| | | |
− | ===JPEG=== | + | ===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> |
| | | |
− | {| align="center" border="1" cellpadding="6" | + | : <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" | | | | valign="bottom" | |
− | <p>[[Image:Rooted Node Big.jpg|20px]]</p><br> | + | <p>[[Image:Rote 1 Big.jpg|20px]]</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:Rote 2 Big.jpg|40px]]</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:Rote 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:Rote 4 Big.jpg|65px]]</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:Rote 5 Big.jpg|40px]]</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:Rote 6 Big.jpg|80px]]</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) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 7 Big.jpg|65px]]</p><br> | | <p>[[Image:Rote 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:Rote 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> |
| + | <p><math>a(8) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 9 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!2 \\ 9 \end{array}</math></p> | + | <p><math>\text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(9) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 10 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 10 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(10) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 11 Big.jpg|40px]]</p><br> | | <p>[[Image:Rote 11 Big.jpg|40px]]</p><br> |
− | <p><math>\begin{array}{l} 5\!:\!1 \\ 11 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}_{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(11) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 12 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 12 Big.jpg|105px]]</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> |
| + | <p><math>a(12) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 13 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 13 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 6\!:\!1 \\ 13 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p} \text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(13) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 14 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 14 Big.jpg|105px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(14) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 15 Big.jpg|80px]]</p><br> | | <p>[[Image:Rote 15 Big.jpg|80px]]</p><br> |
− | <p><math>\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}</math></p> | + | <p><math>\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!</math></p><br> |
| + | <p><math>a(15) ~=~ 3</math></p> |
| |- | | |- |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 16 Big.jpg|90px]]</p><br> | | <p>[[Image:Rote 16 Big.jpg|90px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!4 \\ 16 \end{array}</math></p> | + | <p><math>\text{p}^{\text{p}^\text{p}}\!</math></p><br> |
| + | <p><math>a(16) ~=~ 3</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 17 Big.jpg|65px]]</p><br> | | <p>[[Image:Rote 17 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 7\!:\!1 \\ 17 \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" | | | | valign="bottom" | |
| <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> | | <p>[[Image:Rote 18 Big.jpg|120px]]</p><br> |
− | <p><math>\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}</math></p> | + | <p><math>\text{p} \text{p}_\text{p}^\text{p}\!</math></p><br> |
| + | <p><math>a(18) ~=~ 2</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 19 Big.jpg|65px]]</p><br> | | <p>[[Image:Rote 19 Big.jpg|65px]]</p><br> |
− | <p><math>\begin{array}{l} 8\!:\!1 \\ 19 \end{array}</math></p> | + | <p><math>\text{p}_{\text{p}^{\text{p}_\text{p}}}\!</math></p><br> |
| + | <p><math>a(19) ~=~ 4</math></p> |
| | valign="bottom" | | | | valign="bottom" | |
| <p>[[Image:Rote 20 Big.jpg|105px]]</p><br> | | <p>[[Image:Rote 20 Big.jpg|105px]]</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}_\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 2,692: |
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> |
| | | |