A078304 Generalized Fermat numbers: 7^(2^n)+1, n >= 0.

8, 50, 2402, 5764802, 33232930569602, 1104427674243920646305299202, 1219760487635835700138573862562971820755615294131238402

Offset: 0

Author

Eric W. Weisstein, Nov 21 2002

Keywords

• nonn
• easy

Comments

From Daniel Forgues, Jun 19 2011: (Start)

Generalized Fermat numbers F_n(a) := F_n(a,1) = a^(2^n)+1, a >= 2, n >= 0, can't be prime if a is odd (as is the case for the current sequence) (Ribenboim (1996)).

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, 1998)). (This only expresses that the factors are odd, which means that it only applies to odd generalized Fermat numbers.) (End)

Examples

			a(0) = 7^1+1 = 8 = 6*(1)+2 = 6*(empty product)+2.
a(1) = 7^2+1 = 50 = 6*(8)+2.
a(2) = 7^4+1 = 2402 = 6*(8*50)+2.
a(3) = 7^8+1 = 5764802 = 6*(8*50*2402)+2.
a(4) = 7^16+1 = 33232930569602 = 6*(8*50*2402*5764802)+2.
a(5) = 7^32+1 = 1104427674243920646305299202 = 6*(8*50*2402*5764802*33232930569602)+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.
• 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, A152581, A080176, A199592, A152585.

Programs

Magma

[7^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Mathematica

Table[7^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

Formula

a(0) = 8, a(n)=(a(n-1)-1)^2+1, n >= 1.

a(n) = 6*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 6*(empty product, i.e., 1)+ 2 = 8 = a(0). This means that the GCD of any pair of terms is 2. - Daniel Forgues, Jun 20 2011

Sum_{n>=0} 2^n/a(n) = 1/6. - Amiram Eldar, Oct 03 2022

Extensions

Edited by Daniel Forgues, Jun 19 2011