bibkey: mathar2026a073759
authors: R. J. Mathar
year: 2026
title: "OEIS A073759: largest number neither a divisor of nor relatively prime to n"
doi: null
url: https://oeis.org/A073759
claim: "Conjecture: a(n) = 2*A046667(n) if n>=5. - _R. J. Mathar_, May 15 2026"
strata_touched:
  - D5/S3/Arith/Primes/MatharA073759
license: citation-only
triage: anchor

Mathar’s A073759 conjecture

Verified locator

Source: https://oeis.org/A073759

The NAME and FORMULA fields are unpaginated. The companion definitions occur in the unpaginated FORMULA fields of https://oeis.org/A046666 and https://oeis.org/A046667.

Source statements

A073759, NAME: “Largest number < n that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.” Offset 1.

A073759, FORMULA: “Conjecture: a(n) = 2*A046667(n) if n>=5. - R. J. Mathar, May 15 2026”.

A046666, FORMULA: “a(n) = n - (smallest prime dividing n).” Offset 2.

A046667, FORMULA: “a(n) = A046666(n)/2.” Offset 2.

The candidates are positive integers below n. The Lean definition uses the supremum of their filtered finite interval, with zero for the empty set. Nat.minFac gives the least prime factor for n at least two, and Nat.div is division in the natural numbers. The theorem proves the conjecture for every n at least five. Its proof identifies the maximum as n minus its least prime factor and then uses evenness of this difference.

Evidence and scope

Preregistration: https://github.com/the-omega-institute/trureturing/issues/18937. The source screen reported no cited resolution in these three entries; completeness of the literature search is ASSUMED-UNVERIFIED.

Finite corroboration is the experiment docs/reports/oeis-a073759-mathar-largest-unrelated/ in https://github.com/the-omega-institute/trureturing-experiments at commit 04bafcea983207a1fa6056376aa68d57319d2fa1. The check.py SHA-256 is f36bac2eba413b9af6d1f92f1a3f6ad6ff06bdd62fd4fd3613797bf5fc7633f0. The command python3 check.py exited 0 with RESULT PASS, agreeing for 5 through 20000 and with the first 50 listed terms. It also gives a(4)=0 while 4 minus its least prime factor is 2. These finite checks do not supply the universal proof.