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Claim. P. Pollack and L. Troupe, Sums of proper divisors follow the Erdős--Kac law, Proc. Amer. Math. Soc. 151 (2023), no. 3, 977--988, Theorem 1: for each fixed real $u$, as $x\to\infty$,

$$\frac1x\#\{1<n\le x:\omega(s(n))-\log\log x\le u\sqrt{\log\log x}\} \to\frac1{\sqrt{2\pi}}\int_{-\infty}^{u}e^{-t^2/2}\,dt$$

(card).

Derived as the card states, not stated as a theorem in the paper: for every $h(m)\to\infty$ the target $\{m\ge3:|\omega(m)-\log\log m|>h(m)(\log\log m)^{1/2}\}$ has density zero by the classical Erdős--Kac theorem, and Theorem 1, with $\log\log x$ replaced by $\log\log s(n)$ for all but $o(x)$ of the $n\le x$, gives it a density-zero preimage, the assertion of Problem 955 for that target.

Covers. For every $h(m)\to\infty$, the target $\{m\ge3:|\omega(m)-\log\log m|>h(m)(\log\log m)^{1/2}\}$; the general assertion stays open.

Depends on. No page of this wiki.

Acceptance. Refereed: Proceedings of the American Mathematical Society, a journal. The site's commentary does not credit the result, and the site labels the problem OPEN, so no curator acceptance is listed.