A080174 -id:A080174 - cp's OEIS frontend

A080176 Generalized Fermat numbers: 10^(2^n) + 1, n >= 0.

11, 101, 10001, 100000001, 10000000000000001, 100000000000000000000000000000001, 10000000000000000000000000000000000000000000000000000000000000001

Offset: 0

Jens Voß, Feb 04 2003

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).

			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.

Vincenzo Librandi, Table of n, a(n) for n = 0..9 (shortened by _N. J. A. Sloane_, Jan 13 2019)
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.
Romeo Meštrović, 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.

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

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

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

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

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
Sum_{n>=0} 2^n/a(n) = 1/9. - Amiram Eldar, Oct 03 2022

Edited by Daniel Forgues, Jun 19 2011

A104115 Primes of the form 6^n+7.

Original entry on oeis.org

13, 43, 223, 1303, 46663, 21936950640377863, 4738381338321616903, 1023490369077469249543, 221073919720733357899783, 1047532535594334222593508922191671036215303

Offset: 1

Roger L. Bagula, Mar 05 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..16

Cf. A080174.

[ a: n in [0..250] | IsPrime(a) where a is 6^n+7]; // Vincenzo Librandi, Jul 19 2012

Select[Table[6^n+7,{n,0,300}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

A097547 a(n) = n^(2^n).

Original entry on oeis.org

0, 1, 16, 6561, 4294967296, 23283064365386962890625, 63340286662973277706162286946811886609896461828096, 1487815647197611695910312681741273570332356717154798949898498305086387315423300999654757561928633305897036801

Offset: 0

Yasutoshi Kohmoto, Aug 28 2004

nonn

Number of mappings from 2^X to X where X is an n-set.

Vincenzo Librandi, Table of n, a(n) for n = 0..9 (shortened by _N. J. A. Sloane_, Jan 13 2019)
Yasutoshi Kohmoto, The Other Side of Mathematics [broken link?]
Yasutoshi Kohmoto, The Other Side of Mathematics [via Internet Archive Wayback-machine]

Cf. A000290, A020955, A099129, A216992.

[n^(2^n): n in [0..12]]; // Vincenzo Librandi, May 25 2011

a:= n-> (t-> (t@@n)(n))(j-> j^2):
seq(a(n), n=0..7);  # Alois P. Heinz, Sep 29 2023

Table[n^2^n,{n,0,7}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2010 *)

a(n)=n^(1<Charles R Greathouse IV, May 25 2011

From R. J. Mathar, Apr 23 2007: (Start)
a(6) = A080174(6) - 1.
a(5) = A013737(10) = A013835(6).
a(4) = A000079(32) = A002489(4) = A000215(5)-1.
(End)
Sum_{n>=1} 1/a(n) = A216992. - Amiram Eldar, Nov 19 2020

A104116 Primes of the form 6^n+13.

Original entry on oeis.org

19, 229, 7789, 279949, 28430288029929701389, 174588755932389037098918153698611839369229, 24071951487285782880390320326214662876142414926450873355734389735932734339553143491252967333529911309

Offset: 1

Roger L. Bagula, Mar 05 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..12

Cf. A080174.

[ a: n in [0..250] | IsPrime(a) where a is 6^n+13]; // Vincenzo Librandi, Jul 19 2012

Select[Table[6^n+13,{n,0,300}],PrimeQ] (* Vincenzo Librandi, Jul 19 2012 *)

Corrected a(7) by - Vincenzo Librandi, Jul 19 2012

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A080176 Generalized Fermat numbers: 10^(2^n) + 1, n >= 0.

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A104115 Primes of the form 6^n+7.

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A097547 a(n) = n^(2^n).

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A104116 Primes of the form 6^n+13.

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