cp's OEIS Frontend

A078901 Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>1.

%I A078901 #11 Jul 23 2025 07:48:46
%S A078901 17,97,257,337,881,1921,3697,6497,6817,10657,16561,24641,35377,49297,
%T A078901 65537,66977,72097,89041,116161,149057,188497,235297,290321,354481,
%U A078901 428737,456161,514097,611617,722401,847601,988417,1146097,1321937
%N A078901 Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>1.
%C A078901 It can be shown that, like the Fermat numbers, two of these generalized Fermat numbers are coprime if they have the same base k. However, unlike the Fermat numbers (which are conjectured to be squarefree), these generalized Fermat numbers are not necessarily squarefree for k > 1. Riesel tabulates some prime factors of generalized Fermat numbers for k <= 5.
%C A078901 For k=1, these are the Fermat numbers A000215. See A078900 for the case m>0, which includes the sum of consecutive squares. By Legendre's theorem (Riesel, p. 165), the prime factors of a generalized Fermat number are of the form 1 + f 2^(m+1) for some integer f. See A078902 for generalized Fermat primes.
%D A078901 H. Riesel, "Prime numbers and computer methods for factorization," Second Edition, Progress in Mathematics, Vol. 126, Birkhauser, Boston, 1994, pp. 417-425.
%H A078901 T. D. Noe, <a href="http://www.sspectra.com/math/GenFermat.txt">Factorizations of Generalized Fermat Numbers</a>
%H A078901 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>
%t A078901 mx=10^7; maxK=Ceiling[Sqrt[mx/2]]; maxM=Ceiling[Log[2, Log[2, mx]]]; lst={}; Do[gf=(k+1)^2^m+k^2^m; If[gf<mx, AppendTo[lst, gf]], {k, maxK}, {m, 2, maxM}]; lst2=Union[lst]
%Y A078901 Cf. A000215, A078900, A078902.
%K A078901 easy,nonn
%O A078901 1,1
%A A078901 _T. D. Noe_, Dec 12 2002
%E A078901 Offset corrected by _Sean A. Irvine_, Jul 23 2025