bibkey: sloane2011a008365
authors: N. J. A. Sloane; Gary Detlefs
year: 2011
title: "OEIS A008365, 13-rough numbers: positive integers that have no prime factors less than 13"
doi: null
url: https://oeis.org/A008365
claim: "A008365 %N: 13-rough numbers: positive integers that have no prime factors less than 13. A008365 %C: Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - _Gary Detlefs_, Dec 30 2011"
strata_touched:
  - D5/S0/Certificates/DetlefsRoughResidueCharacterizationRefutation
license: citation-only
triage: anchor

OEIS A008365

The NAME of A008365 states:

13-rough numbers: positive integers that have no prime factors less than 13.

Gary Detlefs’s COMMENT of December 30, 2011 states:

Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - Gary Detlefs, Dec 30 2011

The AUTHOR line names:

N. J. A. Sloane

The AUTHOR line is undated; 2011 is the datable year of Detlefs’s conjecture.

Verified locator

• URL: https://oeis.org/A008365
• NAME (verbatim): 13-rough numbers: positive integers that have no prime factors less than 13.
• COMMENT, Dec 30 2011 (verbatim): Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - Gary Detlefs, Dec 30 2011
• AUTHOR (verbatim): N. J. A. Sloane

The four-residue characterization is refuted by the prime 17, since 17^24 is congruent to 1681 modulo 2310. The corrected five-residue set {1,421,631,841,1681} is disclosed, not claimed: because 2310 = 235711, a value coprime to 2310 has twenty-fourth power congruent to 1 modulo 235*7, while modulo 11 its twenty-fourth power equals its fourth power and can take the five quartic residues {1,3,4,5,9}; 1681 is congruent to 9 modulo 11.

A bounded scan through 200000 found 8311 mismatches, all in the 13-rough-but-outside-four-set direction and none in the four-set-but-not-rough direction; the least mismatches were 17, 61, 71, 83, 127, 137, 149, 181, 193, and 247. In that scan, the 13-rough values realized exactly the residues {1,421,631,841,1681}; these bounded observations are not proofs of the corrected characterization or its converse.