**Edit [2023 Dec 7]:** One of my specific wonders, along with that of students, is around *when* a recursive formula might have – or be expected to have – an explicit or closed formula. What is the mathematical intuition or relevant theorem around the existence of such formulae, and how would that apply to this specific example?

The following question concerns a natural generalization of the following problem and, as relevant, a wonder around when a recursive formula can/not be written in an explicit form:

In exploring this problem, (secondary school) students and I generated enough examples of prime structures to find the following, relevant paper:

Miller, Michael D. "A Recursively Defined Divisor Function."

The Fibonacci Quarterly, 13(3), pages 199-204. Accessible athttps://www.fq.math.ca/Scanned/13-3/miller.pdf.

Define $\gamma: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ by $\gamma(1) = 1$ and, for $N > 1$, let $\gamma(N)$ be the total number of ones in $N$'s factor family.

Question:For $N$ with prime factorization $\prod_i p_i^{\alpha_i}$, what is an explicit formula for $\gamma(N)$?Ifan explicit formula cannot be written for this function,thenwhat is the barrier – articulated "rigorously" or heuristically – preventing it from being written as such?

**Theorem 4** in the linked paper gives a recursive method for computing $\gamma(N)$, but, in the paper's own words:

This theorem, although giving much information about the nature of the function $\gamma$, does not explicitly give us a formula from which we can calculate $\gamma(N)$ for various values of $N$ (page 201).

Hence the question above. Re-tagging and pointers to other references are all welcome!