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

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

Offset

0,1

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

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.

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

Example

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

Mathematica

Table[8^2^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) [8^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011
(PARI) a(n)=1<<(3*2^n)+1 \\ Charles R Greathouse IV, Jul 29 2011

Crossrefs

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

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

Sequence in context: A100311 A259242 A120286 * A122733 A118465 A279129

Adjacent sequences: A152578 A152579 A152580 * A152582 A152583 A152584

Keyword

nonn,easy

Author

Cino Hilliard, Dec 08 2008

Extensions

Edited by Daniel Forgues, Jun 19 2011

Status

approved