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

1,1

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

Links

Table of n, a(n) for n=1..24.

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 *)

Crossrefs

Cf. A001220, A001567, A374841, A001567.

Sequence in context: A043591 A022256 A237925 * A251057 A256002 A225560

Adjacent sequences: A374950 A374951 A374952 * A374954 A374955 A374956

Keyword

nonn

Author

Thomas Ordowski, Jul 25 2024

Extensions

More terms from Amiram Eldar, Jul 25 2024

Status

approved