T. Fink

Three new closed-form expressions give the number of recursive divisors and ordered factorisations, which were until now hard to compute.

The number of recursive divisors and the number of ordered factorisations are two related arithmetic sequences that are recursively defined. But their recursive nature makes it hard to compute explicit values. We give three new closed-form expressions for these quantities, one of which is a generalised hypergeometric function. They shed light on the structure of these sequences and their number-theoretic properties.

The number of ordered factorizations and the number of recursive divisors are two related arithmetic sequences that are recursively defined. But their recursive nature makes it hard to compute explicit values. We provide three closed-form representations of these functions, one of which is, suprisingly, a simple generalized hypergeometric function. They shed light on the structure of these functions and their number-theoretic properties.

Counting recursive divisors

Number of ordered factorizations and recursive divisors