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bibkey: ferreol2018a239293 authors: Robert Ferreol; Thomas Ordowski year: 2018 title: “OEIS A239293, Smallest composite c > n such that n^c == n (mod c)” doi: null url: https://oeis.org/A239293 claim: “%N Smallest composite c > n such that n^c == n (mod c). Context %C: a(n) is the smallest weak pseudoprime to base n that is > n. Context %C: If n is even and n+1 is composite, then a(n) = n+1. [Corrected by Thomas Ordowski, Aug 03 2018] Target %C: Conjecture: a(n) = n+1 if and only if n+1 is an odd composite number. - Thomas Ordowski, Aug 03 2018” strata_touched:

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

OEIS A239293

The NAME of A239293 states:

Smallest composite c > n such that n^c == n (mod c).

The neighbouring COMMENT gives the weak-pseudoprime terminology:

a(n) is the smallest weak pseudoprime to base n that is > n.

The next COMMENT records a sufficient condition:

If n is even and n+1 is composite, then a(n) = n+1. [Corrected by Thomas Ordowski, Aug 03 2018]

Thomas Ordowski’s target COMMENT states:

Conjecture: a(n) = n+1 if and only if n+1 is an odd composite number. - Thomas Ordowski, Aug 03 2018

The AUTHOR line states:

Robert FERREOL, Mar 14 2014

Robert Ferreol is the sequence author, while Thomas Ordowski authored the target conjecture. The bibliographic year 2018 records the conjecture date, not the 2014 sequence-author date.

The formal result settles the target biconditional by classifying the immediate successor as a qualifying composite exactly when it is odd. It does not claim that the sequence value exists for every natural base.

Verified locator

  • URL: https://oeis.org/A239293
  • NAME (verbatim): Smallest composite c > n such that n^c == n (mod c).
  • COMMENT (verbatim): a(n) is the smallest weak pseudoprime to base n that is > n.
  • COMMENT (verbatim): If n is even and n+1 is composite, then a(n) = n+1. [Corrected by Thomas Ordowski, Aug 03 2018]
  • COMMENT (verbatim): Conjecture: a(n) = n+1 if and only if n+1 is an odd composite number. - Thomas Ordowski, Aug 03 2018
  • AUTHOR (verbatim): Robert FERREOL, Mar 14 2014

At the immediate successor, minimality is automatic: a(n)=n+1 is equivalent to n+1 itself satisfying the composite and congruence conditions. The result proves that this happens exactly for odd composite n+1; it does not prove the totality of a.