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Source. Theorem 3.4, 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 $s(m)=\sigma(m)-m$, and for a positive integer $k$, $G(k)$ is the number of pairs of primes $p>q$ with $k=p+q$ (p. 6).

Theorem 3.4 (p. 7). For $n>1$, the number of integers $m$ with $(m,n)=1$ and $s(m)=n$ is

$$G(n-1)+O\bigl(n^{3/4}\log n\bigr).$$

The main term comes from squarefree $m=pq$ with $p>q$, for which $s(pq)=n$ is the same as $p+q+1=n$ (p. 7).

Proof pointer

Proof on pp. 7--8, by cases on $\omega(m)$. The cases $\omega(m)\le1$ give $O(\log n)$ choices; $\omega(m)=2$ gives $G(n-1)$ squarefree choices and $O(n^{1/2}\log n)$ others; $\omega(m)=3$ gives $O(n^{3/4})$ squarefree choices, by counting roots of $x^2+x+n$ modulo $l=s(qr)$, and $O(n^{3/4}/\log n)$ others through Lemma 3.1 (p. 6). The case $\omega(m)\ge4$ uses Proposition 3.5 (p. 8), which splits $m=uv$ with $(u,v)=1$, $v\le n^{3/4}$, and either $u<v$ or $\omega(u)=1$, together with Lemma 3.1 in the case $D=1$.

Dependencies

Lemmas 3.1 and 3.2 (pp. 6--7) and Proposition 3.5 (p. 8) of the paper. 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.