A255925 Bases b for which exactly four Wieferich primes p with p < b exist such that p is a base-b Wieferich prime.

116, 117, 118, 233, 245, 249, 251, 261, 269, 276, 298, 325, 369, 374, 401, 423, 460, 485, 487, 505, 526, 604, 618, 629, 653, 717, 721, 723, 737, 776, 793, 838, 851, 856, 857, 863, 867, 881, 893, 932, 962, 969, 978, 1025, 1037, 1045, 1057, 1059, 1079, 1106

Offset: 1

Author

Felix Fröhlich, Mar 23 2015

Keywords

• nonn

Comments

Numbers b such that A255920(b) = 4.

Links

• Felix Fröhlich, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Wieferich Prime

Crossrefs

Cf. A255920.

Cf. bases b with exactly k base-b Wieferich primes less than b: A255921 (k=0), A255922 (k=1), A255923 (k=2), A255924 (k=3), A325881 (k=5), A325882 (k=6), A325883 (k=7), A325884 (k=8), A325885 (k=9), A325886 (k=10).

Programs

Mathematica

wp[b_] := Count[Complement[Prime[Range[PrimePi[b]]], FactorInteger[b][[All, 1]] ], p_ /; Divisible[b^(p - 1) - 1, p^2]];
Select[Range[2, 1200], wp[#] == 4&] (* Jean-François Alcover, Nov 26 2017 *)

PARI

is(n) = my(i=0); forprime(p=1, n-1, if(Mod(n, p^2)^(p-1)==1, i++); if(i > 4, return(0))); i==4

Sage

[b for b in range(3,1107) if len([p for p in range(2,b) if is_prime(p) and mod(b, p^2)^(p-1)==1])==4] # Danny Rorabaugh, Mar 31 2015