A126032 -id:A126032 - cp's OEIS frontend

A078303 Generalized Fermat numbers: 6^(2^n) + 1, n >= 0.

7, 37, 1297, 1679617, 2821109907457, 7958661109946400884391937, 63340286662973277706162286946811886609896461828097

Offset: 0

Eric W. Weisstein, Nov 21 2002

The next term is too large to include.
As for standard Fermat numbers 2^(2^n) + 1, a number (2b)^m + 1 (with b > 1) can only be prime if m is a power of 2. On the other hand, out of the first 13 base-6 Fermat numbers, only the first three are primes.
Either the sequence of (standard) Fermat numbers contains infinitely many composite numbers or the sequence of base-6 Fermat numbers contains infinitely many composite numbers (cf. https://mathoverflow.net/a/404235/1593). - José Hernández, Nov 09 2021
Since all powers of 6 are congruent to 6 (mod 10), all terms of this sequence are congruent to 7 (mod 10). - Daniel Forgues, Jun 22 2011
There are only 5 known Fermat primes of the form 2^(2^n) + 1: {3, 5, 17, 257, 65537}. There are only 2 known base-10 generalized Fermat primes of the form 10^(2^n) + 1: {11, 101}. - Alexander Adamchuk, Mar 17 2007

			a(0) = 6^1+1 = 7 = 5*(1)+2 = 5*(empty product)+2;
a(1) = 6^2+1 = 37 = 5*(7)+2;
a(2) = 6^4+1 = 1297 = 5*(7*37)+2;
a(3) = 6^8+1 = 1679617 = 5*(7*37*1297)+2;
a(4) = 6^16+1 = 2821109907457 = 5*(7*37*1297*1679617)+2;
a(5) = 6^32+1 = 7958661109946400884391937 = 5*(7*37*1297*1679617*2821109907457)+2;

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.
C. K. Caldwell, "Top Twenty" page, Generalized Fermat Divisors (base=6).
Wilfrid Keller, GFN06 factoring status.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.
OEIS Wiki, Generalized Fermat numbers.

Cf. A000215 (Fermat numbers: 2^(2^n) + 1, n >= 0).
Cf. A019434 (Fermat primes of the form 2^(2^n) + 1).
Cf. A123669, A123599, A056993, A126032, A178428, A059919, A199591, A078304, A152581, A080176, A199592, A152585.

[6^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

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

a(n)=6^(2^n)+1 \\ Charles R Greathouse IV, Jun 21 2011

a(0) = 7, a(n) = (a(n-1)-1)^2 + 1, n >= 1.
a(n) = 5*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 5*(empty product, i.e., 1)+ 2 = 7 = a(0). This implies that the terms are pairwise coprime. - Daniel Forgues, Jun 20 2011
Sum_{n>=0} 2^n/a(n) = 1/5. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 22 2011

A228101 a(n) is the least k such that (2n)^(2^k) + 1 is a prime; a(n) = -1 if a prime (2n)^(2^k) + 1 is unknown, or = -2 if impossible.

Original entry on oeis.org

0, 0, 0, -2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, -2, 2, 0, -1, 0, 0, 4, 0, 2, -1, 0, 1, 1, 0, 0, -1, -2, 0, -1, 0, 0, 1, 4, 0, 2, 0, 1, -1, 0, 1, -1, 1, 0, -1, 0, 0, -1, 0, 0, 1, 0, 5, 1, 2, 1, -1, 1, 0, -2, 0, 2, 1, 0, 0, 2, 2, -1, 1, 0, 0, 3, 2, 0, 4, 1, 0, 2, 0

Offset: 1

Yves Gallot (galloty(AT)wanadoo.fr) and Robert G. Wilson v, Aug 14 2013

A prime number of the form b^(2^k) + 1 is called a generalized Fermat prime to base b.
See the hyperlink for more information and links.
The impossibility case, a(n) = -2, occurs exactly if 2n is a member of A070265. Or equivalently, n is in A126032. - Jeppe Stig Nielsen, Jul 02 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..500
Lucile and Yves Gallot, Welcome to the Generalized Fermat Prime Search!

Cf. A056993, A079706, A084712, A070265, A126032.

f[b_?EvenQ] := f[b] = Block[{k = 0}, While[! PrimeQ[b^(2^k) + 1], k++]; k];
lst = {38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998}; (f[#] = -1) & /@ lst;
lst = {8, 32, 64, 128, 216, 512, 1000}; (f[#] = -2) & /@ lst; Table[ f[b], {b, 2, 1000, 2}]
(* Second program: *)
Module[{r = 83, nn = 12, s = {}, k}, Do[If[b > r, Break[], Do[If[Set[k, b^m/2] > r, Break[], AppendTo[s, k]], {m, 3, Infinity, 2}]], {b, 2, Infinity, 2}]; Table[If[MemberQ[s, n], -2, SelectFirst[Range[0, nn], PrimeQ[(2 n)^(2^#) + 1] &] /. x_ /; MissingQ@ x -> -1], {n, r}]] (* Michael De Vlieger, Jul 04 2017, Version 10.2 *)

Definition rewritten by Jeppe Stig Nielsen, Jul 02 2017

cp's OEIS Frontend

A078303 Generalized Fermat numbers: 6^(2^n) + 1, n >= 0.

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