login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A078304 Generalized Fermat numbers: 7^(2^n)+1, n >= 0. 13
8, 50, 2402, 5764802, 33232930569602, 1104427674243920646305299202, 1219760487635835700138573862562971820755615294131238402 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

From Daniel Forgues, Jun 19 2011: (Start)

Generalized Fermat numbers 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 the current sequence) (Ribenboim (1996)).

All factors of generalized Fermat numbers F_n(a,b) := a^(2^n)+b^(2^n), a >= 2, n >= 0, are of the form k*2^m+1, k >= 1, m >=0 (Riesel (1994, 1998)). (This only expresses that the factors are odd, which means that it only applies to odd generalized Fermat numbers.) (End)

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.

Eric Weisstein's World of Mathematics, Generalized Fermat Number

OEIS Wiki, Generalized Fermat numbers

FORMULA

a(0) = 8, a(n)=(a(n-1)-1)^2+1, n >= 1.

a(n) = 6*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 6*(empty product, i.e., 1)+ 2 = 8 = a(0). This means that the GCD of any pair of terms is 2. - Daniel Forgues, Jun 20 2011

EXAMPLE

a(0) = 7^1+1 = 8 = 6*(1)+2 = 6*(empty product)+2.

a(1) = 7^2+1 = 50 = 6*(8)+2.

a(2) = 7^4+1 = 2402 = 6*(8*50)+2.

a(3) = 7^8+1 = 5764802 = 6*(8*50*2402)+2.

a(4) = 7^16+1 = 33232930569602 = 6*(8*50*2402*5764802)+2.

a(5) = 7^32+1 = 1104427674243920646305299202 = 6*(8*50*2402*5764802*33232930569602)+2.

MATHEMATICA

Table[7^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

PROG

(MAGMA) [7^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

CROSSREFS

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

Cf. A059919, A199591, A078303, A152581, A080176, A199592, A152585.

Sequence in context: A195231 A162236 A215874 * A000851 A054620 A034516

Adjacent sequences:  A078301 A078302 A078303 * A078305 A078306 A078307

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein, Nov 21 2002

EXTENSIONS

Edited by Daniel Forgues, Jun 19 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 15 05:43 EDT 2019. Contains 328026 sequences. (Running on oeis4.)