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Directory:Jon Awbrey/Papers/Riffs and Rotes (view source)
Revision as of 19:26, 3 February 2010
, 19:26, 3 February 2010→Idea
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where <math>\text{p}_{i(k)}^{j(k)}</math> is the <math>k^\text{th}</math> prime power in the factorization and <math>\ell</math> is the number of distinct prime factors dividing <math>n.</math>
where <math>\text{p}_{i(k)}^{j(k)}</math> is the <math>k^\text{th}</math> prime power in the factorization and <math>\ell</math> is the number of distinct prime factors dividing <math>n.</math> The factorization of <math>1</math> is defined as <math>1</math> in accord with the convention that an empty product is equal to <math>1.</math>
Let <math>I(n)</math> be the set of indices of primes that divide <math>n</math> and let <math>j(i, n)</math> be the number of times that <math>\text{p}_i</math> divides <math>n.</math> Then the prime factorization of <math>n</math> can be written in the following alternative form:
Let <math>I(n)</math> be the set of indices of primes that divide <math>n</math> and let <math>j(i, n)</math> be the number of times that <math>\text{p}_i</math> divides <math>n.</math> Then the prime factorization of <math>n</math> can be written in the following alternative form:
9876543210
9876543210
& = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721
& = & 2 \cdot 3^2 \cdot 5 \cdot {17}^2 \cdot 379721
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1
& = & \text{p}_1^1 \text{p}_2^2 \text{p}_3^1 \text{p}_7^2 \text{p}_{32277}^1.
\end{matrix}</math>
\end{matrix}</math>
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Each index <math>i</math> and exponent <math>j</math> appearing in the prime factorization of a positive integer <math>n</math> is itself a positive integer, and thus has a prime factorization of its own.
==Riffs in Numerical Order==
==Riffs in Numerical Order==