Claims

../

2014_05_14_troupe: Troupe proves that omega(s(n)) and Omega(s(n)) are within eps log log s(n) of log log s(n) for all but o(x) integers n up to x, so the integers with abnormally many or few prime factors have a density-zero preimage.

2015_04_01_pollack: Pollack proves that s(n) is prime for only O(x/log x) integers n up to x, so the preimage of the primes under s has density zero.

2015_06_15_pollack: Pollack proves that for each base g at least 2, s(n) is a base-g palindrome only for a density-zero set of n, the conjecture for the palindromes.

2017_06_09_pollack_pomerance_thompson: Pollack, Pomerance and Thompson prove that a set of at most x to the one-half plus o(1) integers has o(x) preimages up to x under s, settling the conjecture for every target of counting function y^(1/2+o(1)).

2019_02_28_troupe: Troupe proves that s(n) is a sum of two squares for a number of n up to x of order x over the square root of log x, so the sums of two squares have a density-zero preimage under s.

2021_06_20_pollack_troupe: Pollack and Troupe prove that omega(s(n)) satisfies the Erdős-Kac law, so the integers whose prime-factor count is far from log log m in Erdős-Kac units have a density-zero preimage.

2023_07_24_benli_cesana_dartyge_dombrowsky_thompson: Benli, Cesana, Dartyge, Dombrowsky and Thompson prove that the n up to x with every base-g digit of s(n) in a fixed proper digit set number O(x exp(-(log log x)^gamma)) for g at least 3.

2026_07_21_benli_dartyge_dombrowsky_pollack_thompson: Benli, Dartyge, Dombrowsky, Pollack and Thompson extend the missing-digit preimage bound to every base g at least 2, supplying the binary case in an appendix.

---