---

Source. Theorem 1.1, p. 2 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

Notation (pp. 1--2). $s(n)=\sigma(n)-n$ is the sum of the proper divisors of $n$, extended by $s(0)=0$, and $s_k$ is the $k$-th iterate of $s$. Bosma and Kane proved that there is a real number $\beta$ with

$$\frac1x\sum_{n\le x}\log\bigl(s(2n)/2n\bigr)\to\beta\qquad(x\to\infty),$$

and the paper records $\beta\approx-0.03$ (p. 2).

Theorem 1.1 (p. 2). As $x\to\infty$,

$$\frac1x\sum_{2\le n\le x}\log\bigl(s_2(2n)/s(2n)\bigr)\;\sim\; \frac1x\sum_{1\le n\le x}\log\bigl(s(2n)/2n\bigr)\;\sim\;\beta .$$

The first sum starts at $n=2$ because $s_2(2)=0$ (p. 2). The theorem concerns even arguments $2n$ only and one further step of the iteration; it says nothing about the growth of an individual aliquot sequence.

Proof pointer

Section 2 (pp. 3--5). A density-one statement $s_2(n)/s(n)\sim s(n)/n$ from earlier work is not enough, since a density-zero set of large terms could move the average, so the proof controls the large terms. Large negative values of $\log(s_2(2n)/s(2n))$ are rare because $s_2(2n)/s(2n)<1/2$ forces $s(2n)$ odd, so $n$ or $2n$ is a square. Large positive values of $s(2n)/2n$ are handled by Theorem E (p. 3, an upper bound for the number of $n\le x$ with $s(n)/n>y$, attributed to Erdős). The core is Proposition 2.1 (p. 4): for all but $O(x/y^{4/3})$ integers $n\le x$, with $y=(\log_2x)/(\log_3x)^2$, $|s_2(n)/s(n)-s(n)/n|\ll(\log_4x/\log_3x)\cdot\sigma(n)/n$.

Dependencies

The Bosma–Kane theorem (the paper's reference [3]) and Theorem E (the paper's reference [14, Theorem B]), both cited, not proved, in the paper. Read depth: claims checked; the statement was read clause by clause on p. 2 and the proof for its structure only.

Bears on

• Problem 410: background only. The problem concerns the iterates of $\sigma$, while the theorem concerns the average of one step of the iteration of $s=\sigma-\mathrm{id}$ over even arguments; it gives no statement about $\sigma_k(n)^{1/k}$.