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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 primes 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>

Let <math>I(n)</math> be the set of indices of primes that divide the positive integer <math>n.</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>j(i, n)</math> be the number of times that <math>\text{p}_i</math> divides <math>n.</math>

 

The prime factorization of a positive integer <math>n</math> can be written in the following form:

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