A059919 Generalized Fermat numbers: 3^(2^n)+1, n >= 0.
4, 10, 82, 6562, 43046722, 1853020188851842, 3433683820292512484657849089282, 11790184577738583171520872861412518665678211592275841109096962
Offset: 0
Examples
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;
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.
Crossrefs
Programs
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Magma
[3^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011
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Maple
A059919:=n->3^(2^n)+1; seq(A059919(n), n=0..7); # Wesley Ivan Hurt, Jan 22 2014
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Mathematica
Table[3^2^n + 1, {n, 0, 7}] (* Arkadiusz Wesolowski, Nov 02 2012 *) -
PARI
a(n) = { 3^(2^n) + 1 } \\ Harry J. Smith, Jun 30 2009
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
Extensions
Edited by Daniel Forgues, Jun 19 2011 and Jun 20 2011
Comments