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

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

Offset

0,1

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 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.

References

H. Riesel, "Prime numbers and computer methods for factorization," Second Edition, Progress in Mathematics, Vol. 126, Birkhauser, Boston, 1994, pp. 417-425.

Links

Table of n, a(n) for n=0..32.

T. D. Noe, Factorizations of Generalized Fermat Numbers

Eric Weisstein's World of Mathematics, Generalized Fermat Number

Mathematica

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]

Crossrefs

Cf. A000215, A078900, A078902.

Sequence in context: A325068 A264823 A081593 * A078902 A103766 A165347

Adjacent sequences: A078898 A078899 A078900 * A078902 A078903 A078904

Keyword

easy,nonn

Author

T. D. Noe, Dec 12 2002

Status

approved