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A078303 Generalized Fermat numbers: 6^(2^n) + 1, n >= 0. 13
7, 37, 1297, 1679617, 2821109907457, 7958661109946400884391937, 63340286662973277706162286946811886609896461828097 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

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

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

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.

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

FORMULA

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

EXAMPLE

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;

MATHEMATICA

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

PROG

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

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

CROSSREFS

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.

Sequence in context: A292807 A210620 A250843 * A127729 A129736 A220852

Adjacent sequences:  A078300 A078301 A078302 * A078304 A078305 A078306

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein, Nov 21 2002

EXTENSIONS

Edited by Daniel Forgues, Jun 22 2011

STATUS

approved

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Last modified August 25 03:10 EDT 2019. Contains 326318 sequences. (Running on oeis4.)