A374953 Numbers k such that 2^(2^k-2) == 1 (mod k^2) and 2^(k-1) =/= 1 (mod k).

66709, 951481, 2215441, 2847421, 4111381, 4869757, 28758601, 81844921, 124187581, 300510001, 306197821, 1221936841, 9763146541, 10370479321, 13560714361, 14387344201, 16287076081, 16956342901, 18820810297, 19245374461, 22732640101, 26946809137, 27119213281, 29217386881

Offset: 1

Author

Thomas Ordowski, Jul 25 2024

Keywords

• nonn

Comments

The composite terms of A374841 that are not in A001567.

Every term of this sequence must have a Wieferich prime factor (for example 66709 = 19 * 3511). Wieferich prime p = 1093 cannot divide such k, since it would require ord_{p^2}(2) = 364 = 2^2 * 91 to divide 2^k - 2, which is impossible. - Max Alekseyev, Jul 25 2024

Crossrefs

Cf. A001220, A001567, A374841, A001567.

Programs

Mathematica

f[n_] := Module[{e = IntegerExponent[n^2, 2], d, k, r}, d = n^2 / 2^e; k = MultiplicativeOrder[2, d]; r = PowerMod[2, n, k] - e - 2; r = Mod[r, k]; 2^e * PowerMod[2, r, d]];
q[n_] := PowerMod[2, n-1, n] != 1 && f[n] == 1;
Select[Range[10^6], CompositeQ[#] && q[#] &] (* or: *)
Select[3511 * Range[10^5], q] (* faster, can be used for generating terms up to 4.97*10^17, the current lower bound for A001220(3) *)
(* Amiram Eldar, Jul 25 2024 after T. D. Noe at A155836 *)

Extensions

More terms from Amiram Eldar, Jul 25 2024