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

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

Offset

0,1

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)

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.

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A059919, A199591, A078303, A152581, A080176, A199592, A152585.

Sequence in context: A162236 A215874 A348594 * A000851 A054620 A034516

Adjacent sequences: A078301 A078302 A078303 * A078305 A078306 A078307

Keyword

nonn,easy

Author

Eric W. Weisstein, Nov 21 2002

Extensions

Edited by Daniel Forgues, Jun 19 2011

Status

approved