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bibkey: lava2010a119690 authors: Paolo P. Lava; Giorgio Balzarotti; John W. Layman year: 2010 title: “OEIS A119690, n! mod n*(n+1)/2” doi: null url: https://oeis.org/A119690 claim: “%N n! mod n*(n+1)/2. Target %C: It appears that f(n)=(n!)^(2k+1) modulo n(n+1)/2 is n if n is one less than an odd prime, else f(n) is 0, for any integer k. See A175567 for related results involving an even power of n!. - John W. Layman, Jul 12 2010. %F a(n) = n if n+1 is an odd prime, a(n) = 0 otherwise.” strata_touched:

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

OEIS A119690

The NAME of A119690 states:

n! mod n*(n+1)/2.

John W. Layman’s target COMMENT states:

It appears that f(n)=(n!)^(2k+1) modulo n(n+1)/2 is n if n is one less than an odd prime, else f(n) is 0, for any integer k. See A175567 for related results involving an even power of n!. - John W. Layman, Jul 12 2010

The FORMULA line states:

a(n) = n if n+1 is an odd prime, a(n) = 0 otherwise.

The AUTHOR line states:

Paolo P. Lava and Giorgio Balzarotti, Jun 09 2006

Paolo P. Lava and Giorgio Balzarotti authored the sequence, while John W. Layman authored the target conjectural comment. The bibliographic year 2010 records the conjecture date, not the 2006 sequence-author date.

The formal result settles only Layman’s generalization to all odd powers. The case k = 0 is already the entry’s own FORMULA, and no priority is claimed for that case.

Verified locator

  • URL: https://oeis.org/A119690
  • NAME (%N, verbatim): n! mod n*(n+1)/2.
  • COMMENT (%C, verbatim): It appears that f(n)=(n!)^(2k+1) modulo n(n+1)/2 is n if n is one less than an odd prime, else f(n) is 0, for any integer k. See A175567 for related results involving an even power of n!. - John W. Layman, Jul 12 2010
  • FORMULA (%F, verbatim): a(n) = n if n+1 is an odd prime, a(n) = 0 otherwise.
  • AUTHOR (%A, verbatim): Paolo P. Lava and Giorgio Balzarotti, Jun 09 2006

Only the all-odd-powers generalization in Layman’s COMMENT is settled here. The k = 0 case is the entry’s own FORMULA, and no priority is claimed for that case.