Closed formula for number of ones in a proper factor tree

Question

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?

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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 at https://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 $\gamma(N)$? If an explicit formula cannot be written for this function, then what 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!

Answer 1

This seems to be Sloane's A002033, namely the number of perfect partitions of $n-1$. Since the OEIS doesn't give any closed formula, there probably isn't one, but it's probably worth checking the references pointed to there.

Answer 2

Take the formal product $g(x_1,x_2,\ldots)=\prod_{i\ge 1} (1-x_i)$ and define $$f(x_1,x_2,\ldots) = \frac{g(x_1,x_2,\ldots)}{2g(x_1,x_2,\ldots)-1}.$$ Then $\gamma(\prod_i p_i^{\alpha_i})$ is the coefficient of $\prod_i x_i^{\alpha_i}$ in the Taylor expansion of $f(x_1,x_2,\ldots)$.

This is elementary and I don't see how it gives an "explicit" formula except in simple cases.

Answer 3

A simple generating function, though not a closed formula, for $\gamma(N)$ is given by $\sum_{N\geq 1}\frac{\gamma(N)}{N^s} = \frac{1}{2-\zeta(s)}$, where $\zeta(s)$ is the Riemann zeta function. See e.g. page 7 of E. C. Titchmarsh, The Zeta Function of Riemann.