A129786 Least k such that 2^(2^n)+k is prime.

0, 1, 1, 1, 1, 15, 13, 51, 297, 75, 643, 981, 1761, 897, 2775, 118113, 44061, 5851, 18531, 189093, 69661

Offset: 0

Author

Benoit Cloitre, May 18 2007

Keywords

• nonn
• more

Comments

It is conjectured that a(n)>=3 for n>=5.

For n>11, 2^(2^n)+a(n) is a probable prime. By a comment in A000215, a(n) is not 2^m+1 for any m > 1. - T. D. Noe, Jul 19 2007

Links

• C. K. Caldwell, The Prime Glossary, Fermat prime.
• Eric Weisstein's World of Mathematics, Fermat Number.
• Eric Weisstein's World of Mathematics, Fermat Prime.

Crossrefs

Cf. A000215, A019434.

Cf. A013597 (least k>0 such that 2^n+k is prime).

Programs

Mathematica

a[n_] := Module[{k = 0}, While[! PrimeQ[2^(2^n) + k], k++]; k]; Array[a, 12, 0] (* Amiram Eldar, Jun 11 2022 *)

PARI

a(n)=if(n<0,0,s=0;while(isprime(2^(2^n)+s)==0,s++);s)

Python

from sympy import nextprime
def a(n): m = 2**(2**n); return nextprime(m-1) - m
print([a(n) for n in range(12)]) # Michael S. Branicky, Jun 12 2022

Extensions

More terms from T. D. Noe, Jul 19 2007

a(18)-a(19) by Makoto Morimoto, added by Boyan Hu, Jul 05 2025

a(20) by Boyan Hu, Jul 05 2025