bibkey: zelinskyzhang2025klprimitive authors: Joshua Zelinsky; Kyle Zhang year: 2025 title: “Kullback-Leibler divergence and primitive non-deficient numbers” doi: 10.48550/arXiv.2501.04209 url: https://arxiv.org/abs/2501.04209v2 claim: “Conjecture 22: for every positive integer n outside {1, 12, 24, 30, 36, 48, 60, 72, 120, 180, 240, 360}, if p is the smallest prime factor of n, then v(n) >= 1/p^2, where v(n) is the divisor-weighted logarithmic sum defined in the paper.” strata_touched:

• D5/S0/Certificates/ZelinskyZhangConjectureTwentyTwoRefutation license: citation-only triage: anchor

Kullback-Leibler divergence and primitive non-deficient numbers

Zelinsky and Zhang define, for a positive integer n,

v(n) = sum_{d | n, d > 1} (1/d) log((tau(n) - 1)/d).

On page 14, Conjecture 22 states that every positive integer outside the twelve listed exclusions satisfies v(n) >= 1/p^2, where p is its smallest prime factor.

Verified locator

• DOI: 10.48550/arXiv.2501.04209
• URL: https://arxiv.org/abs/2501.04209v2
• Locator: arXiv:2501.04209v2, page 14, Conjecture 22.

The current arXiv record lists versions v1 and v2. The result formalized in the named stratum refutes the literal universal statement at n = 6.