bibkey: wiseman2019a328959 authors: Gus Wiseman year: 2019 title: “OEIS A328959, a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n)” doi: null url: https://oeis.org/A328959 claim: “a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222. Conjecture: All terms are nonnegative except for a(1) = -1. - Gus Wiseman, Nov 02 2019. The idea for this sequence came from Mats Granvik” strata_touched:

• D5/S3/Arith/Congruence/DivisorCountPrimeFactorBound license: citation-only triage: anchor

OEIS A328959

The entry defines the integer-valued sequence a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 is A000005, nu is A001221, and omega is A001222. Thus the entry’s omega counts prime factors with multiplicity, while its nu counts distinct prime factors. In conventional notation the formula is a(n) = sigma_0(n) - 2 - (Omega(n) - 1) * omega(n).

The conjecture asserts nonnegativity except at n = 1, where a(1) = -1. The formal theorem proves the nonnegative branch for every n >= 2.

Verified locator

• URL: https://oeis.org/A328959
• Locator: COMMENTS, Gus Wiseman, Nov 02 2019; FORMULA.
• COMMENTS (verbatim): “Conjecture: All terms are nonnegative except for a(1) = -1.”
• FORMULA (verbatim): “a(n) = A000005(n) - A307408(n). - Antti Karttunen, Nov 17 2019”