A155836 -id:A155836 - cp's OEIS frontend

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

Thomas Ordowski, Jul 25 2024

nonn

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

Cf. A001220, A001567, A374841, A001567.

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 *)

More terms from Amiram Eldar, Jul 25 2024

A137327 Fermat(n) modulo n.

Original entry on oeis.org

0, 1, 2, 1, 2, 5, 5, 1, 5, 7, 4, 5, 10, 3, 2, 1, 2, 17, 5, 17, 5, 17, 4, 17, 22, 17, 14, 17, 17, 17, 9, 1, 5, 19, 12, 17, 34, 17, 23, 17, 38, 17, 5, 21, 32, 7, 22, 17, 5, 17, 2, 17, 43, 53, 37, 17, 29, 55, 21, 17, 58, 17, 5, 1, 62, 17, 22, 53, 65, 17, 13, 17

Offset: 1

Zerinvary Lajos, Apr 20 2008

nonn

In the first 10000 terms, 17 appears 649 times and 257 appears 781 times. - T. D. Noe, Oct 01 2013

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A155836.

with(numtheory):seq(irem(fermat(n),n),n=1..19);
# second Maple program:
a:= n-> (2&^(2^n) + 1) mod n:
seq(a(n), n=1..100); # Alois P. Heinz, Sep 30 2013

Table[Mod[PowerMod[2, 2^n, n] + 1, n], {n, 100}] (* T. D. Noe, Oct 01 2013 *)

a(n) = A000215(n) mod n.

A155886 Least k such that 2^(2^k) = n (mod k).

Original entry on oeis.org

1, 3, 14, 11, 6, 1941491, 10, 83, 31, 13, 123, 35, 71, 27, 34913, 241, 18, 8059, 34, 349, 44, 25, 39, 100867561, 76, 231, 253, 66203, 57, 227, 139, 45, 184, 37, 111, 97, 55, 41, 103, 1099, 81, 66791, 53

Offset: 0

T. D. Noe, Jan 29 2009

First occurrence of n in sequence A155836.

a(43) > 12500000. - Tyler Busby, Mar 15 2024

Cf. A015910 (2^n mod n), A036236, A155836.

cp's OEIS Frontend

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

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A137327 Fermat(n) modulo n.

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A155886 Least k such that 2^(2^k) = n (mod k).

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