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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A080176 Generalized Fermat numbers: 10^(2^n) + 1, n >= 0. 12
11, 101, 10001, 100000001, 10000000000000001, 100000000000000000000000000000001, 10000000000000000000000000000000000000000000000000000000000000001 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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 12 base-10 Fermat numbers, only the first two are primes.

Also, binary representation of Fermat numbers (in decimal, see A000215).

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=10)

Wilfrid Keller, GFN10 factoring status

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012

Eric Weisstein's World of Mathematics, Generalized Fermat Number

OEIS Wiki, Generalized Fermat numbers

FORMULA

a(0) = 11; a(n) = (a(n - 1) - 1)^2 + 1.

a(n) = 9*a(n-1)*a(n-2)*...*a(1)*a(0) + 2, n >= 0, where for n = 0, we get 9*(empty product, i.e., 1)+ 2 = 11 = a(0). - Daniel Forgues, Jun 20 2011

EXAMPLE

a(0) = 10^1 + 1 = 11 = 9*(1) + 2 = 9*(empty product) + 2.

a(1) = 10^2 + 1 = 101 = 9*(11) + 2.

a(2) = 10^4 + 1 = 10001 = 9*(11*101) + 2.

a(3) = 10^8 + 1 = 100000001 = 9*(11*101*10001) + 2.

a(4) = 10^16 + 1 = 10000000000000001 = 9*(11*101*10001*100000001) + 2.

a(5) = 10^32 + 1 = 100000000000000000000000000000001 = 9*(11*101*10001*100000001*10000000000000001) + 2.

MATHEMATICA

Table[10^2^n + 1, {n, 0, 6}] (* Arkadiusz Wesolowski, Nov 02 2012 *)

PROG

(MAGMA) [10^(2^n) + 1: n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

CROSSREFS

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

Cf. A019434, A080174, A080175, A059919, A199591, A078303, A078304, A152581, A199592, A152585.

Sequence in context: A052075 A070854 A075767 * A064490 A080439 A098153

Adjacent sequences:  A080173 A080174 A080175 * A080177 A080178 A080179

KEYWORD

easy,nonn

AUTHOR

Jens Voß, Feb 04 2003

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 27 07:55 EDT 2017. Contains 288777 sequences.