Search: id:A078900
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%I A078900
%S A078900 5,13,17,25,41,61,85,97,113,145,181,221,257,265,313,337,365,421,481,545,
%T A078900 613,685,761,841,881,925,1013,1105,1201,1301,1405,1513,1625,1741,1861,
%U A078900 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 square-free), 
               these generalized Fermat numbers are not necessarily square-free 
               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="http://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.
%Y A078900 Sequence in context: A089545 A121727 A119321 this_sequence A113482 A077426 
               A002144
%Y A078900 Adjacent sequences: A078897 A078898 A078899 this_sequence A078901 A078902 
               A078903
%K A078900 easy,nonn
%O A078900 0,1
%A A078900 T. D. Noe (noe(AT)sspectra.com), Dec 12 2002

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