A175865 Numbers k with property that 2^(k-1) == 1 (mod k) and 2^((3*k-1)/2) - 2^((k-1)/2) + 1 == 0 (mod k).

3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, 101, 107, 109, 131, 139, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 251, 269, 277, 283, 293, 307, 317, 331, 347, 349, 373, 379, 389, 397, 419, 421, 443, 461, 467, 491, 499, 509, 523, 541, 547, 557, 563

Offset

1,1

Comments

All composites in this sequence are 2-pseudoprimes, see A001567.

The subsequence of composites begins: 3277, 29341, 49141, 80581, 88357, 104653, 196093, 314821, 458989, 476971, 489997, ..., . - Robert G. Wilson v, Oct 02 2010

The sequence includes all the primes of A003629. - Alzhekeyev Ascar M, Mar 09 2011

If we consider the composites in this sequence which are in the modulo classes == 3 (mod 8) or == 5 (mod 8), they are moreover strong pseudoprimes to base 2 (see A001262). - Alzhekeyev Ascar M, Mar 09 2011

Are there any composites in this sequence which are *not* in the two modulo classes == {3,5} (mod 8)? - R. J. Mathar, Mar 29 2011

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

3 is a term since 2^(3-1)-1 = 3 is divisible by 3, and 2^((3*3-1)/2) - 2^((3-1)/2) + 1 = 15 is divisible by 3.

Mathematica

fQ[n_] := PowerMod[2, n - 1, n] == 1 && Mod[ PowerMod[2, (3 n - 1)/2, n] - PowerMod[2, (n - 1)/2, n], n] == n - 1; Select[ Range@ 570, fQ] (* Robert G. Wilson v, Oct 02 2010 *)

Crossrefs

Cf. A001262, A001567, A003629.

Sequence in context: A059646 A319041 A003629 * A001122 A152871 A329760

Adjacent sequences: A175862 A175863 A175864 * A175866 A175867 A175868

Keyword

nonn

Author

Alzhekeyev Ascar M, Sep 30 2010

Extensions

More terms from Robert G. Wilson v, Oct 02 2010

Status

approved