cp's OEIS Frontend

A078900 Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>0.

%I A078900 #11 Jul 23 2025 07:48:58
%S A078900 5,13,17,25,41,61,85,97,113,145,181,221,257,265,313,337,365,421,481,
%T A078900 545,613,685,761,841,881,925,1013,1105,1201,1301,1405,1513,1625,1741,
%U A078900 1861,1921,1985,2113,2245,2381,2521,2665,2813,2965,3121,3281,3445,3613,3697
%N A078900 Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>0.
%C A078900 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 A078900 For k=1, these are the Fermat numbers A000215. See A078901 for the case m>1, which excludes 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.
%D A078900 H. Riesel, "Prime numbers and computer methods for factorization," Second Edition, Progress in Mathematics, Vol. 126, Birkhauser, Boston, 1994, pp. 417-425.
%H A078900 T. D. Noe, <a href="http://www.sspectra.com/math/GenFermat.txt">Factorizations of Generalized Fermat Numbers</a>
%H A078900 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeneralizedFermatNumber.html">Generalized Fermat Number</a>
%t A078900 mx=5000; 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, 1, maxM}]; lst1=Union[lst]
%Y A078900 Cf. A000215, A078901.
%K A078900 easy,nonn
%O A078900 1,1
%A A078900 _T. D. Noe_, Dec 12 2002
%E A078900 Offset corrected by _Sean A. Irvine_, Jul 23 2025