bibkey: krizek2015a265714
authors: Jaroslav Krizek; Robert Israel
year: 2015
title: OEIS A265714, minima of reciprocal-sigma divisor sums
doi: null
url: https://oeis.org/history/view?seq=A265714&v=23
claim: For each positive integer threshold, the least positive argument whose reciprocal-sigma divisor sum reaches the threshold also has floor equal to the threshold.
strata_touched:
  - D5/S3/Factorization/ReciprocalSigmaThreshold
license: citation-only
triage: anchor

Reciprocal-sigma threshold minima

The definition in revision 23 is “a(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.” Its conjecture says that a(n) is also the smallest number k for which the sum is at least n. The index starts at one. Here sigma(d) is the sum of all positive divisors of d. Both sets must be nonempty for every positive index, with no upper bound on index or positive argument, including index one.

Robert Israel’s comments give multiplicativity, the prime value 1 + 1/(p+1), divergence of the associated prime product, and existence of every level. These are prior source observations, not new claims of the repository proof. The prime-power expansion is an elementary consequence of the divisor-sum definition; revision 23 does not state that formula explicitly. The separate question about integral values above one and the proposed numerical values at indices four through seven are outside this assertion.

Verified locator

url: https://oeis.org/history/view?seq=A265714&v=23

The primary text identifies revision 23 at 2022-09-08 08:46:15. Its SHA-256 is 39a670f78590ad0fdc102fbd1c9d4a3daeea8cc9d461dfd7ea9189d4ffc95c2e. A conjecture label records what this source says; it does not establish worldwide absence of a subsequent proof.