A126032 Numbers of the form b^m/2 for even b and odd m > 2.

4, 16, 32, 64, 108, 256, 500, 512, 864, 1024, 1372, 2048, 2916, 3888, 4000, 4096, 5324, 6912, 8192, 8788, 10976, 13500, 16384, 19652, 23328, 27436, 32000, 37044, 42592, 48668, 50000, 55296, 62500, 65536, 70304, 78732, 87808, 97556, 108000, 119164, 124416

Offset: 1

Author

Alexander Adamchuk, Feb 28 2007

Keywords

• easy
• nonn

Comments

The old definition was: Numbers n such that A123669(n) = -1, or no generalized Fermat prime exists of the form (2n)^(2^k) + 1. But that sequence is probably missing a lot of terms, such as {6, 9, 11, 19, 21, 25, 26, 29, 30, 31, ...}, where no generalized Fermat prime has been found yet, and it seems unlikely any exist. Currently it can only be proved that none exist if n is of form b^m/2 for even b and odd m > 1. The listed terms are the first numbers of this form: 4 = 2^3/2, 16 = 2^5/2, 32 = 4^3/2, 64 = 2^7/2, 108 = 6^3/2, 256 = 2^9/2 = 8^3/2, 500 = 10^3/2. - Jens Kruse Andersen, Jul 24 2014

The even terms of A070265, divided by two. - Jeppe Stig Nielsen, Jul 02 2017

Links

• Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
• Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Crossrefs

Cf. A123669 = Smallest generalized Fermat prime of the form (2n)^(2^k) + 1, where k>0.

Cf. A070265.

Programs

Mathematica

Module[{nn = 2^17, a = {}, n}, Do[If[b > nn, Break[], Do[If[Set[n, b^m/2] > nn, Break[], AppendTo[a, n]], {m, 3, Infinity, 2}]], {b, 2, Infinity, 2}]; Union@ a] (* Michael De Vlieger, Jul 04 2017 *)

PARI

isOK(n)=ip=ispower(2*n);ip&&bitand(ip,ip-1) \\ Jeppe Stig Nielsen, Jul 02 2017

Extensions

Definition changed by N. J. A. Sloane, Jul 26 2014 following the advice of Jens Kruse Andersen.

Terms after a(7) from Jeppe Stig Nielsen, Jul 02 2017