cp's OEIS Frontend

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

%I A126032 #17 Feb 16 2025 08:33:04
%S A126032 4,16,32,64,108,256,500,512,864,1024,1372,2048,2916,3888,4000,4096,
%T A126032 5324,6912,8192,8788,10976,13500,16384,19652,23328,27436,32000,37044,
%U A126032 42592,48668,50000,55296,62500,65536,70304,78732,87808,97556,108000,119164,124416
%N A126032 Numbers of the form b^m/2 for even b and odd m > 2.
%C A126032 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
%C A126032 The even terms of A070265, divided by two. - _Jeppe Stig Nielsen_, Jul 02 2017
%H A126032 Jeppe Stig Nielsen, <a href="http://jeppesn.dk/generalized-fermat.html">Generalized Fermat Primes sorted by base</a>.
%H A126032 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>.
%t A126032 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 *)
%o A126032 (PARI) isOK(n)=ip=ispower(2*n);ip&&bitand(ip,ip-1) \\ _Jeppe Stig Nielsen_, Jul 02 2017
%Y A126032 Cf. A123669 = Smallest generalized Fermat prime of the form (2n)^(2^k) + 1, where k>0.
%Y A126032 Cf. A070265.
%K A126032 easy,nonn
%O A126032 1,1
%A A126032 _Alexander Adamchuk_, Feb 28 2007
%E A126032 Definition changed by _N. J. A. Sloane_, Jul 26 2014 following the advice of _Jens Kruse Andersen_.
%E A126032 Terms after a(7) from _Jeppe Stig Nielsen_, Jul 02 2017