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Search: id:A078901
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| A078901 |
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Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>1. |
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3
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| 17, 97, 257, 337, 881, 1921, 3697, 6497, 6817, 10657, 16561, 24641, 35377, 49297, 65537, 66977, 72097, 89041, 116161, 149057, 188497, 235297, 290321, 354481, 428737, 456161, 514097, 611617, 722401, 847601, 988417, 1146097, 1321937
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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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.
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.
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REFERENCES
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H. Riesel, "Prime numbers and computer methods for factorization," Second Edition, Progress in Mathematics, Vol. 126, Birkhauser, Boston, 1994, pp. 417-425.
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LINKS
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T. D. Noe, Factorizations of Generalized Fermat Numbers
Eric Weisstein's World of Mathematics, Generalized Fermat Number
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A000215, A078900, A078902.
Adjacent sequences: A078898 A078899 A078900 this_sequence A078902 A078903 A078904
Sequence in context: A070186 A142189 A081593 this_sequence A078902 A103766 A008514
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KEYWORD
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easy,nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Dec 12 2002
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