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
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Dec 08 2008
EXTENSIONS
Edited by Daniel Forgues, Jun 19 2011
STATUS
approved