bibkey: vospost2004a100952
authors: Jonathan Vos Post; Reinhard Zumkeller
year: 2004
title: "OEIS A100952, Numbers that cannot be written as p*q+r with three distinct primes p, q and r"
doi: null
url: https://oeis.org/A100952
claim: "A100952 %N: Numbers that cannot be written as p*q+r with three distinct primes p, q and r. A100952 %C: A weaker conjecture: every integer greater than 60 (or some larger value based on further search) may be partitioned into a prime p and a semiprime qr, where the prime p is bounded by log(min(q,r)). Chen (1978) showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes. Zumkeller's conjecture effectively extends this from \"even\" to both even and odd integers. - _Jonathan Vos Post_, Nov 25 2004 A100952 %A: _Reinhard Zumkeller_, Nov 23 2004"
strata_touched:
  - D5/S3/Arith/Congruence/VosPostLogBoundedSemiprimePartitionRefutation
license: citation-only
triage: anchor

OEIS A100952

The NAME of A100952 defines the exceptional-number sequence:

Numbers that cannot be written as p*q+r with three distinct primes p, q and r.

The COMMENT line records a weaker logarithmically bounded conjecture. The repository result refutes only that sub-conjecture: every integer congruent to 5 modulo 6 fails, so no threshold works. The completeness conjecture for the sequence is untouched.

Verified locator

• URL: https://oeis.org/A100952
• NAME (verbatim): Numbers that cannot be written as p*q+r with three distinct primes p, q and r.
• COMMENT (verbatim): A weaker conjecture: every integer greater than 60 (or some larger value based on further search) may be partitioned into a prime p and a semiprime qr, where the prime p is bounded by log(min(q,r)). Chen (1978) showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes. Zumkeller’s conjecture effectively extends this from “even” to both even and odd integers. - Jonathan Vos Post, Nov 25 2004
• AUTHOR (verbatim): Reinhard Zumkeller, Nov 23 2004