A376253 Composite numbers k such that 2^(2^(k-1)-1) == 1 (mod k^2).

4681, 15841, 42799, 52633, 220729, 647089, 951481, 1082401, 1145257, 1969417, 2215441, 3567481, 4835209, 5049001, 5681809, 6140161, 6334351, 8725753, 10712857, 11777599, 12327121, 13500313, 14709241, 22564081, 22849481, 22953673, 23828017, 27271151, 28758601, 30576151

Offset: 1

Author

Thomas Ordowski, Sep 17 2024

Keywords

• nonn

Comments

If 2^(k-1) == 1 (mod k) and 2^(2^(k-1)-1) == 1 (mod k), then 2^(2^(k-1)-1) == 1 (mod k^2). In fact, all such pseudoprimes are strong pseudoprimes to base 2.

Other terms; 951481 = 271*3511, 2215441 = 631*3511, 28758601 = 8191*3511, ... are not Fermat pseudoprimes to base 2, where 3511 is a Wieferich prime. The Wieferich prime 1093 cannot be a factor of these numbers (see A374953).

Crossrefs

Cf. A001220, A001262, A001567, A188465, A374841, A374953.

Programs

Mathematica

q[k_] := Module[{m = MultiplicativeOrder[2, k^2]}, PowerMod[2, k-1, m] == 1]; Select[Range[1, 10^6, 2], CompositeQ[#] && q[#] &] (* Amiram Eldar, Sep 17 2024 *)

PARI

is(k) = (k > 1) && k % 2 && !isprime(k) && Mod(2, znorder(Mod(2, k^2)))^(k-1) == 1; \\ Amiram Eldar, Sep 17 2024

Extensions

More terms from Amiram Eldar, Sep 17 2024