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Claim. L. Troupe, On the number of prime factors of values of the sum-of-proper-divisors function, J. Number Theory 150 (2015), 120--135, Theorem 1.3: for any $\epsilon>0$ and all $n\le x$ outside a set of size $o(x)$,

$$|\omega(s(n))-\log\log s(n)|<\epsilon\log\log s(n),$$

and the paper proves the same for $\Omega(s(n))$ (card). The paper notes that Theorem 1.3 would follow from the conjecture of Problem 955 and proves it unconditionally.

Covers. For each $\epsilon>0$, the density-zero target $\{m:|\omega(m)-\log\log m|\ge\epsilon\log\log m\}$ and its $\Omega$ analogue, which contain the integers with unusually many prime factors that the site names; the general assertion stays open.

Depends on. No page of this wiki.

Acceptance. Refereed: Journal of Number Theory. The site's commentary credits the result, but the site labels the problem OPEN, so no curator acceptance is listed.