A084712 Smallest prime of the form (2n)^k + 1, or 0 if no such number exists.

3, 5, 7, 0, 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, 0, 1336337, 37

Offset: 1

Author

Amarnath Murthy, Jun 10 2003

Keywords

• nonn
• more

Comments

It has not been proved that a(19), a(25), a(31), a(34), a(43) and a(46) are 0; if these values do exist, they have > 4000 digits. The other zeros are definite. - David Wasserman, Jan 03 2005

a((p-1)/2) = p for primes p > 2, or a(n) = 2n+1 for n = (p-1)/2. All other positive a(n) belong to A002496 = primes of form m^2 + 1. Corresponding positive exponents k are powers of 2. They are listed in A079706. - Alexander Adamchuk, Sep 17 2006

Because k must be a power of 2, numbers of the form (2n)^k+1 are called generalized Fermat numbers with base 2n. These numbers, like the regular Fermat numbers, are seldom prime. I checked n=19, 25, 31, 34, 43, 46 with k up to 2^16 without finding any primes. - T. D. Noe, May 13 2008

Comments from N. J. A. Sloane, Jan 27 2024: (Start)

As pointed out by Max Alekseyev, the previous version violated the OEIS rules, since a(19) has not been confirmed. I therefore removed the terms starting at a(19).

The previous DATA line read:

3, 5, 7, 0, 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, 0, 1336337, 37, 0, 41, 43, 197352587024076973231046657, 47, 5308417, 0, 53, 2917, 3137, 59, 61, 0, 0, 67, 0, 71, 73, 5477, 1238846438084943599707227160577, 79, 40960001, 83, 7057, 0, 89

The old b-file has been changed to an a-file.

(End)

Examples

			a(7) = 197 = 14^2 + 1 as 14 + 1 = 15 is not a prime.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..500 (currently known terms, may contain 0s in place of unknown terms)

Crossrefs

Cf. A084713, A005097.

Programs

Mathematica

Table[k=1; While[p=1+(2n)^k; k<1024 && !PrimeQ[p], k=2k]; If[k==1024, 0, p], {n,44}] (* T. D. Noe, May 13 2008 *)

Extensions

More terms from David Wasserman, Jan 03 2005

Edited by N. J. A. Sloane, Jan 27 2024 at the suggestion of Max Alekseyev