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

2,1

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A138908 A032564 A141248 * A088899 A258199 A290205

Adjacent sequences: A273336 A273337 A273338 * A273340 A273341 A273342

Keyword

nonn

Author

Felix Fröhlich, May 20 2016

Status

approved