Doubly Recursive Integer Factorizations
The fundamental theorem of arithmetic states that every integer greater than 1 can be factored uniquely (up to the ordering of the factors) into a product of powers of one or more primes. So denoting the primes 2, 3, 5 … as , , , …, each integer can be factored as , where and are the index and power of the prime in the factorization. For example, , or .[more]
After that, each index and exponent can likewise be factored: . Because the recursive process ends with all indexes and exponents as 1, the ones can be suppressed without loss of meaning, so that .
This Demonstration shows this representation of any integer factorization.[less]
Snapshot 1: the integer 2 is the first prime raised to the first power, with representation
Snapshot 2: subscript and superscript are ones displayed
Snapshot 3: hiding the subscript and superscript ones changes the order of factors due to the sorting rules in Mathematica.
 "Riffs and Rotes." (Aug 7, 2018) oeis.org/wiki/Riffs_and_Rotes.