A376473 Numbers k such that 2^(2^(k-1)-1) == 1 (mod k^2) and 2^(k-1) =/= 1 (mod k).

951481, 2215441, 28758601, 81844921, 1221936841, 10370479321, 16287076081, 26946809137, 33663998161, 35094800881, 134619011281, 188455112353, 299226038833, 314240366881, 383116075201, 594981050401, 1230227375833, 1572186445201, 2096189123113, 2377714473001

Offset: 1

Author

Thomas Ordowski, Sep 24 2024

Keywords

• nonn

Comments

The terms k of A374953 for which A002326((k-1)/2) is odd.

Numbers k in A376253 that are not strong pseudoprimes to base 2.

Every term of this sequence must have a Wieferich prime factor (for example, 951481 = 271 * 3511). The Wieferich prime 1093 cannot divide such a number (see A374953).

Crossrefs

Subsequence of A374953.

Cf. A001220, A002326, A001262, A001567, A376253.

Programs

Mathematica

q[k_] := Module[{m = MultiplicativeOrder[2, k^2]}, PowerMod[2, k - 1, m] == 1]; Select[Range[1, 2300000, 2], PowerMod[2, # - 1, #] != 1 && q[#] &] (* Amiram Eldar, Sep 24 2024 *)

PARI

is(k) = (k > 1) && k % 2 && !isprime(k) && Mod(2, k)^(k-1) != 1 && Mod(2, znorder(Mod(2, k^2)))^(k-1) == 1; \\ Amiram Eldar, Sep 24 2024

PARI

list(lim)=my(v=List()); if(lim>3<<64, warning("May miss multiples of Wieferich primes > 2^64.")); forstep(n=10533,lim,7022, if(Mod(2, znorder(Mod(2, n^2)))^(n-1) == 1 && Mod(2,n)^n != 2, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Sep 24 2024

Extensions

More terms from Amiram Eldar, Sep 24 2024