A178428 -id:A178428 - 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

A178427 7 followed by the Fermat numbers A152581.

Original entry on oeis.org

7, 9, 65, 4097, 16777217, 281474976710657, 79228162514264337593543950337, 6277101735386680763835789423207666416102355444464034512897

Offset: 0

Roger L. Bagula, May 27 2010

If a(0)=3, the recursion formula gives A000215.

Cf. A000215, A152581, A178426, A178428.

a[0] := 7;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] + 2;
Table[a[n], {n, 0, 10}]

a(0)=7. a(n) = 2 + Product_{i=0..n-1} a(i).

Definition simplified by the Assoc. Eds. of the OEIS - May 28 2010

Incorrect a(8) removed by Georg Fischer, May 22 2024

A178426 11 followed by the Fermat numbers A152585.

Original entry on oeis.org

11, 13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137

Offset: 0

Roger L. Bagula, May 27 2010

If a(0)=3, the recursion formula gives A000215.

Cf. A000215, A178427, A178428.

a[0] := 11;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] + 2;
Table[a[n], {n, 0, 10}]

a(0)=11. a(n)= 2 + Product_{i=0..n-1} a(i).

Definition simplified by the Assoc. Eds. of the OEIS - May 28 2010

A178497 5 followed by A145503.

Original entry on oeis.org

5, 3, 13, 193, 37633, 1416317953, 2005956546822746113, 4023861667741036022825635656102100993, 16191462721115671781777559070120513664958590125499158514329308740975788033

Offset: 0

Roger L. Bagula, May 28 2010

nonn

The first 6 entries of the sequence are primes.

The Fermat numbers F(.)=A000215(.) obey equations with flipped signs of the +-2 if compared with the a(n) (see the formula section): F(n) = Prod{i=0,...,n-1} F(i)+2 and F(n) = F(n-1) *(F(n-1)-2)+2. - Vladimir Shevelev, Dec 08 2010

Cf. A000215, A110407, A178428, A178427.

Clear[a, n];
a[ -1] := 1; a[0] := 5;
a[n_] := a[n] = Product[a[i], {i, 0, n - 1}] - 2;
Table[a[n], {n, 0, 10}]

a(n)= ( product_{i=0..n-1} a(i)) -2, n>=1.

a(n) = a(n-1)*( a(n-1)+2 )-2, n>=2. - Vladimir Shevelev, Dec 08 2010

Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010

cp's OEIS Frontend

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

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A178427 7 followed by the Fermat numbers A152581.

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A178426 11 followed by the Fermat numbers A152585.

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A178497 5 followed by A145503.

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