A152585 Generalized Fermat numbers: 12^(2^n) + 1, n >= 0.

13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137

Offset

0,1

Comments

There appears to be no divisibility rule for this sequence.

13 is the only prime up to 12^(2^15)+1.

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.

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

Example

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.

Mathematica

Table[12^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

Prog

(PARI) g(a, n) = if(a%2, b=2, b=1); for(x=0, n, y=a^(2^x)+b; print1(y", "))
(Magma) [12^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011
(Python)
def A152585(n): return (1<<2*(m:=1<<n))*3**m+1 # Chai Wah Wu, Jul 19 2022

Crossrefs

Cf. A000215 (Fermat numbers: 2^(2^n)+1, n >= 0).

Cf. A059919, A199591, A078303, A078304, A152581, A080176, A199592.

Sequence in context: A270579 A297223 A199023 * A014881 A048442 A353107

Adjacent sequences: A152582 A152583 A152584 * A152586 A152587 A152588

Keyword

easy,nonn

Author

Cino Hilliard, Dec 08 2008

Extensions

Edited by Daniel Forgues, Jun 19 2011

Status

approved