---

Source. Theorem 3.3, p. 7 of the author's manuscript, of Carl Pomerance, The first function and its iterates, in Connections in Discrete Mathematics, Cambridge University Press (2018), 125--138, as identified on the source card. Page numbers are those of the manuscript.

Statement

Here $s(m)=\sigma(m)-m$ is the sum of the proper divisors of $m$.

Theorem 3.3 (p. 7). For a fixed integer $n>1$, the number of integers $m$ with $s(m)=n$ and $(m,n)>1$ is $O_\epsilon(n^{2/3+\epsilon})$ for each $\epsilon>0$.

The implied constant depends only on $\epsilon$. Every such $m$ satisfies $m<n^2$ (p. 7).

Proof pointer

Proof on p. 7. Each such $m$ is written $m=m_0D$ with $1<D<n^2$, $\operatorname{rad}(D)\mid n$ and $(m_0,Dn)=1$. The case $m_0=1$ gives one choice, and Lemma 3.1 (p. 6) counts the choices of $m_0$ when $m_0$ is a prime power or a product of two prime powers; when $\omega(m_0)\ge3$, Lemma 3.2 (p. 7) splits $m_0=uv$ with coprime $u<v<n^{2/3}$, after which the second part of Lemma 3.1 leaves at most $n^{2/3}$ choices. The number of $D<n^2$ with $\operatorname{rad}(D)\mid n$ is $n^{o(1)}$, by results the paper cites from its references [10] and [20].

Dependencies

Lemmas 3.1 and 3.2 (pp. 6--7) of the paper and the cited count of $D$. Read depth: claims checked; the statement was read clause by clause on p. 7 and the proof for its structure only.

Bears on

No Erdős problem page in the corpus is about this count. It feeds Corollary 3.6.