A152581 Generalized Fermat numbers: a(n) = 8^(2^n) + 1, n >= 0.

9, 65, 4097, 16777217, 281474976710657, 79228162514264337593543950337, 6277101735386680763835789423207666416102355444464034512897

Offset: 0

Author

Cino Hilliard, Dec 08 2008

Keywords

• nonn
• easy

Comments

These numbers are all composite. We rewrite 8^(2^n) + 1 = (2^(2^n))^3 + 1.

Then by the identity a^n + b^n = (a+b)*(a^(n-1) - a^(n-2)*b + ... + b^(n-1)) for odd n, 2^(2^n) + 1 divides 8^(2^n) + 1. All factors of generalized Fermat numbers F_n(a,b) := a^(2^n)+b^(2^n), a >= 2, n >= 0, are of the form k*2^m+1, k >= 1, m >=0 (Riesel (1994)). - Daniel Forgues, Jun 19 2011

Examples

			For n = 3, 8^(2^3) + 1 = 16777217. Similarly, (2^8)^3 + 1 = 16777217. Then 2^8 + 1 = 257 and 16777217/257 = 65281.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10
• Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.
• Eric Weisstein's World of Mathematics, Generalized Fermat Number.
• OEIS Wiki, Generalized Fermat numbers.

Crossrefs

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

Cf. A059919, A199591, A078303, A078304, A080176, A199592, A152585.

Programs

Magma

[8^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Mathematica

Table[8^2^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","))

PARI

a(n)=1<<(3*2^n)+1 \\ Charles R Greathouse IV, Jul 29 2011

Formula

a(0)=9, a(n) = (a(n-1) - 1)^2 + 1, n >= 1.

Sum_{n>=0} 2^n/a(n) = 1/7. - Amiram Eldar, Oct 03 2022

Extensions

Edited by Daniel Forgues, Jun 19 2011