A059919 Generalized Fermat numbers: 3^(2^n)+1, n >= 0.

4, 10, 82, 6562, 43046722, 1853020188851842, 3433683820292512484657849089282, 11790184577738583171520872861412518665678211592275841109096962

Offset

0,1

Comments

Generalized Fermat numbers (Ribenboim (1996))

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 this sequence). - Daniel Forgues, Jun 19-20 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10 (shortened by N. J. A. Sloane, Jan 13 2019)

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=3).

Wilfrid Keller, GFN3 factoring status.

Eric Weisstein's World of Mathematics, Generalized Fermat Number.

OEIS Wiki, Generalized Fermat numbers.

Formula

a(0) = 4; a(n) = (a(n-1)-1)^2 + 1, n >= 1.

a(n) = A011764(n)+1 = A059918(n+1)/A059918(n) = (A059917(n+1)-1)/(A059917(n)-1) = (A059723(n)/A059723(n+1))*(A059723(n+2)-A059723(n+1))/(A059723(n+1)-A059723(n))

a(n) = A057727(n)-1. - R. J. Mathar, Apr 23 2007

a(n) = 2*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 2*(empty product, i.e., 1) + 2 = 4 = a(0).

The above formula implies the GCD of any pair of terms is 2, which means that the terms of (3^(2^n)+1)/2 (A059917) are pairwise coprime. - Daniel Forgues, Jun 20 & 22 2011

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

Example

a(0) = 3^(2^0)+1 = 3^1+1 = 4 = 2*(1)+2 = 2*(empty product)+2;
a(1) = 3^(2^1)+1 = 3^2+1 = 10 = 2*(4)+2;
a(2) = 3^(2^2)+1 = 3^4+1 = 82 = 2*(4*10)+2;
a(3) = 3^(2^3)+1 = 3^8+1 = 6562 = 2*(4*10*82)+2;
a(4) = 3^(2^4)+1 = 3^16+1 = 43046722 = 2*(4*10*82*6562)+2;
a(5) = 3^(2^5)+1 = 3^32+1 = 1853020188851842 = 2*(4*10*82*6562*43046722)+2;

Maple

A059919:=n->3^(2^n)+1; seq(A059919(n), n=0..7); # Wesley Ivan Hurt, Jan 22 2014

Mathematica

Table[3^2^n + 1, {n, 0, 7}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

Prog

(PARI) { for (n=0, 11, write("b059919.txt", n, " ", 3^(2^n) + 1); ) } \\ Harry J. Smith, Jun 30 2009
(Magma) [3^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

Crossrefs

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

Cf. A059917 ((3^(2^n)+1)/2).

Cf. A199591, A078303, A078304, A152581, A080176, A199592, A152585.

Sequence in context: A171754 A215872 A371021 * A143047 A156329 A266839

Adjacent sequences: A059916 A059917 A059918 * A059920 A059921 A059922

Keyword

easy,nonn

Author

Henry Bottomley, Feb 08 2001

Extensions

Edited by Daniel Forgues, Jun 19 2011 and Jun 20 2011

Status

approved