cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

7, 37, 1297, 1679617, 2821109907457, 7958661109946400884391937, 63340286662973277706162286946811886609896461828097
Offset: 0

Views

Author

Eric W. Weisstein, Nov 21 2002

Keywords

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.
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

Examples

			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;

Links

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.

Programs

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
Sum_{n>=0} 2^n/a(n) = 1/5. - Amiram Eldar, Oct 03 2022

Extensions

Edited by Daniel Forgues, Jun 22 2011

A126032 Numbers of the form b^m/2 for even b and odd m > 2.

Original entry on oeis.org

4, 16, 32, 64, 108, 256, 500, 512, 864, 1024, 1372, 2048, 2916, 3888, 4000, 4096, 5324, 6912, 8192, 8788, 10976, 13500, 16384, 19652, 23328, 27436, 32000, 37044, 42592, 48668, 50000, 55296, 62500, 65536, 70304, 78732, 87808, 97556, 108000, 119164, 124416
Offset: 1

Views

Author

Alexander Adamchuk, Feb 28 2007

Keywords

Comments

The old definition was: Numbers n such that A123669(n) = -1, or no generalized Fermat prime exists of the form (2n)^(2^k) + 1. But that sequence is probably missing a lot of terms, such as {6, 9, 11, 19, 21, 25, 26, 29, 30, 31, ...}, where no generalized Fermat prime has been found yet, and it seems unlikely any exist. Currently it can only be proved that none exist if n is of form b^m/2 for even b and odd m > 1. The listed terms are the first numbers of this form: 4 = 2^3/2, 16 = 2^5/2, 32 = 4^3/2, 64 = 2^7/2, 108 = 6^3/2, 256 = 2^9/2 = 8^3/2, 500 = 10^3/2. - Jens Kruse Andersen, Jul 24 2014
The even terms of A070265, divided by two. - Jeppe Stig Nielsen, Jul 02 2017

Links

Crossrefs

Cf. A123669 = Smallest generalized Fermat prime of the form (2n)^(2^k) + 1, where k>0.
Cf. A070265.

Programs

  • Mathematica
    Module[{nn = 2^17, a = {}, n}, Do[If[b > nn, Break[], Do[If[Set[n, b^m/2] > nn, Break[], AppendTo[a, n]], {m, 3, Infinity, 2}]], {b, 2, Infinity, 2}]; Union@ a] (* Michael De Vlieger, Jul 04 2017 *)
  • PARI
    isOK(n)=ip=ispower(2*n);ip&&bitand(ip,ip-1) \\ Jeppe Stig Nielsen, Jul 02 2017

Extensions

Definition changed by N. J. A. Sloane, Jul 26 2014 following the advice of Jens Kruse Andersen.
Terms after a(7) from Jeppe Stig Nielsen, Jul 02 2017

A253242 Least k>=0 such that n^(2^k)+1 is prime (for even n), or (n^(2^k)+1)/2 is prime (for odd n); -1 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, -1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 1, 0, 1, -1, 0, 1, 0
Offset: 2

Views

Author

Eric Chen, Apr 19 2015

Keywords

Comments

Least k such that the generalized Fermat number in base n (GFN(k,n)) is prime.
a(n) = -1 if n is in A070265 (perfect powers with an odd exponent).
a(n) is currently unknown for n = {31, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 97, 98, 99, 104, 107, 109, 122, 123, 127, 135, 137, ...}
Corresponding primes are {3, 2, 5, 3, 7, 1201, 0, 5, 11, 61, 13, 7, 197, 113, 17, 41761, 19, 181, 401, 11, 23, 139921, 577, 13, 677, 0, 29, 421, 31, ...}. (use 0 if a(n) = -1)
All 2 <= n <= 1500 and 0 <= k <= 14 are checked, the first occurrence of k (start with k = 0) in a(n) are {2, 11, 7, 43, 41, 75, 274, 234, 331, 1342, 824, ...}.

Examples

			a(7) = 2 since (7^(2^0)+1)/2 and (7^(2^1)+1)/2 are not primes, but (7^(2^2)+1)/2 = 1201 is prime.
a(14) = 1 since 14^(2^0)+1 is not prime, but 14^(2^1)+1 = 197 is prime.

Links

Crossrefs

Cf. A079706, A228101, A084712, A123669, A058064, A057856, A130536, A080121, A077659, A122900.

Programs

  • Mathematica
    Table[k=0; While[p=If[EvenQ[n], (2n)^(2^k)+1, ((2n)^(2^k)+1)/2]; k<12 && !PrimeQ[p], k=k+1]; If[k==12, -1, k], {n, 2, 1500}]
  • PARI
    f(n) = for(k=0, 11, if(ispseudoprime(n^(2^k)+1), return(k))); -1
g(n) = for(k=0, 11, if(ispseudoprime((n^(2^k)+1)/2), return(k))); -1
a(n) = if(n%2==0, f(n), g(n))
  • PARI
    f(n,k)=if(n%2, (n^(2^k)+1)/2, n^(2^k)+1)
a(n)=if(ispower(-n), -1, my(k); while(!ispseudoprime(f(n,k)), k++); k) \\ Charles R Greathouse IV, Apr 20 2015

Formula

a(2n) = A228101(n) = log_2(A079706(n)).
a(A006093(n)) = 0, a(A076274(n)) = 0, a(A070265(n)) = -1.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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