A273339 Smallest composite c such that n^(c-1) != 1 (mod c^2), i.e., smallest composite c that is not a "Wieferich pseudoprime" to base n.

4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4

Offset: 2

Author

Felix Fröhlich, May 20 2016

Keywords

• nonn

Comments

Smallest composite c such that n does not occur in row c of the array in A244752. - Felix Fröhlich, Jan 07 2017

Conjecture: periodic with period = 16. - Harvey P. Dale, May 12 2025

For any set S of composite numbers, if n is a composite == 1 (mod lcm(S)^2) then n^(c-1) == 1 (mod c^2) for all c in S, so a(n) is not in S. Thus the sequence has infinitely many distinct members, and in particular the conjecture is false. - Robert Israel, Aug 15 2025

Links

• Felix Fröhlich, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A240719, A244752, A256517, A267288, A270776.

Programs

Maple

f:= proc(n) local c;
  for c from 4 do
    if not isprime(c) and n &^(c-1) mod (c^2) <> 1 then return c fi
  od
end proc:
map(f, [$2..100]); # Robert Israel, Aug 15 2025

Mathematica

A273339[n_] := NestWhile[#+1 &, 4, PrimeQ[#] || PowerMod[n, #-1, #^2] == 1 &];
Array[A273339, 100, 2] (* Paolo Xausa, Aug 15 2025 *)

PARI

a(n) = forcomposite(c=1, , if(Mod(n, c^2)^(c-1)!=1, return(c)))