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

Author

Alzhekeyev Ascar M, Sep 30 2010

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A001262, A001567, A003629.

Programs

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 *)

Extensions

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