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A078900
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Generalized Fermat numbers of the form (k+1)^2^m + k^2^m, with m>0.
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3
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5, 13, 17, 25, 41, 61, 85, 97, 113, 145, 181, 221, 257, 265, 313, 337, 365, 421, 481, 545, 613, 685, 761, 841, 881, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1921, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3697
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 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.
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.
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REFERENCES
| 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
| T. D. Noe, Factorizations of Generalized Fermat Numbers
Eric Weisstein's World of Mathematics, Generalized Fermat Number
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MATHEMATICA
| 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]
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CROSSREFS
| Cf. A000215, A078901.
Sequence in context: A089545 A121727 A119321 * A113482 A191218 A077426
Adjacent sequences: A078897 A078898 A078899 * A078901 A078902 A078903
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KEYWORD
| easy,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Dec 12 2002
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