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.
%I A084712 #39 Jan 27 2024 13:18:05
%S A084712 3,5,7,0,11,13,197,17,19,401,23,577,677,29,31,0,1336337,37
%N A084712 Smallest prime of the form (2n)^k + 1, or 0 if no such number exists.
%C A084712 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
%C A084712 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
%C A084712 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
%C A084712 Comments from _N. J. A. Sloane_, Jan 27 2024: (Start)
%C A084712 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).
%C A084712 The previous DATA line read:
%C A084712 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
%C A084712 The old b-file has been changed to an a-file.
%C A084712 (End)
%H A084712 Vincenzo Librandi, <a href="/A084712/a084712_1.txt">Table of n, a(n) for n = 1..500</a> (currently known terms, may contain 0s in place of unknown terms)
%e A084712 a(7) = 197 = 14^2 + 1 as 14 + 1 = 15 is not a prime.
%t A084712 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 *)
%Y A084712 Cf. A084713, A005097.
%K A084712 nonn,more
%O A084712 1,1
%A A084712 _Amarnath Murthy_, Jun 10 2003
%E A084712 More terms from _David Wasserman_, Jan 03 2005
%E A084712 Edited by _N. J. A. Sloane_, Jan 27 2024 at the suggestion of _Max Alekseyev_