A152585 Generalized Fermat numbers: 12^(2^n) + 1, n >= 0.
13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137
Offset: 0
Examples
a(0) = 12^1+1 = 13 = 11(1)+2 = 11(empty product)+2. a(1) = 12^2+1 = 145 = 11(13)+2. a(2) = 12^4+1 = 20737 = 11(13*145)+2. a(3) = 12^8+1 = 429981697 = 11(13*145*20737)+2. a(4) = 12^16+1 = 184884258895036417 = 11(13*145*20737*429981697)+2. a(5) = 12^32+1 = 34182189187166852111368841966125057 = 11(13*145*20737*429981697*184884258895036417)+2.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..12
- Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
- C. K. Caldwell, "Top Twenty" page, Generalized Fermat Divisors (base=12).
- Wilfrid Keller, GFN12 factoring status.
- Eric Weisstein's World of Mathematics, Generalized Fermat Number.
- OEIS Wiki, Generalized Fermat numbers.
Crossrefs
Programs
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Magma
[12^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011
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Mathematica
Table[12^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *) -
PARI
g(a,n) = if(a%2,b=2,b=1);for(x=0,n,y=a^(2^x)+b;print1(y","))
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Python
def A152585(n): return (1<<2*(m:=1<
Formula
a(0) = 13; a(n)=(a(n-1)-1)^2 + 1, n >= 1.
a(n) = 11*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 11*(empty product, i.e., 1)+ 2 = 13 = a(0). This implies that the terms, all odd, are pairwise coprime. - Daniel Forgues, Jun 20 2011
Sum_{n>=0} 2^n/a(n) = 1/11. - Amiram Eldar, Oct 03 2022
Extensions
Edited by Daniel Forgues, Jun 19 2011
Comments