A215540 Least k such that (2*n-1)*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m, or 0 if no such value exists.

1, 41, 7, 14, 67, 18759, 20, 229, 147, 6838, 41

Offset: 1

Author

Arkadiusz Wesolowski, Aug 15 2012

Keywords

• nonn
• hard
• more

Comments

(2*n-1)*2^a(n) + 1 is in A023394.

a(n) >= 7 for n > 1.

a(39279) = 0. No n < 39279 with a(n)=0 is known.

a(12)>2500000, a(13)>2500000, a(14)=455, a(15)=57 (see Ballinger and Keller link).

No, a(13)=2141884, found in 2011. - Jeppe Stig Nielsen, Sep 07 2019

Links

• Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
• Fermat factoring status, Prime factors of Fermat numbers
• FermatSearch, Home page
• PrimeGrid, Announcement of 25*2^2141884+1, related to a(13).
• Eric Weisstein's World of Mathematics, Fermat Number

Crossrefs

Cf. A000215, A033809.

Programs

Mathematica

lst = {}; Do[k = 1; While[True, p = n*2^k + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, k]; Break[]]; k++], {n, 1, 9, 2}]; lst