cp's OEIS Frontend

A174246 Number of primes of the form x^2 + 1 < 2^n.

%I A174246 #30 Feb 16 2025 08:33:12
%S A174246 0,1,2,2,3,4,5,6,8,10,12,14,18,24,33,42,54,70,91,114,158,212,293,393,
%T A174246 539,713,957,1301,1792,2459,3378,4615,6233,8418,11540,15867,21729,
%U A174246 29843,41169,56534,77697,106787,147067,203025,280340,387308,535153,739671
%N A174246 Number of primes of the form x^2 + 1 < 2^n.
%C A174246 Terms from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek).
%C A174246 It is conjectured that this sequence is unbounded, but this has never been proved. [Comment corrected by _Kellen Myers_, Oct 12 2014.]
%C A174246 More precisely, it is not known if there are infinitely many primes of the form k^2 + 1. See references and links. - _N. J. A. Sloane_, Oct 14 2014
%C A174246 Same as A083847 except for a(1) = 0. - _Georg Fischer_, Oct 14 2018
%H A174246 Chris Caldwell, <a href="https://primes.utm.edu/notes/conjectures/">Prime Conjectures and Open Questions</a>
%H A174246 G. H. Hardy and J. E. Littlewood, <a href="https://dx.doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
%H A174246 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/LandausProblems.html">Landau's Problems</a>
%e A174246 a(10) = 10 because the only primes or the form x^2 + 1 < 2^10 are the ten primes: 2, 5, 17, 37, 101, 197, 257, 401, 577 & 677.
%p A174246 N:= 30: # to get a(1) to a(N).
%p A174246 P:= select(isprime,[2,seq((2*i)^2+1, i = 1 .. floor(sqrt(2^N-1)/2))]):
%p A174246 seq(nops(select(`<`,P,2^n)), n=1..N); # _Robert Israel_, Oct 13 2014
%o A174246 (PARI) lista(nn) = {nb = 0; for (n=1, nn, forprime(p=2^n, 2^(n+1)-1, if (issquare(p-1), nb++);); print1(nb, ", "););} \\ _Michel Marcus_, Oct 13 2014
%Y A174246 Cf. A083844, A083845, A083846, A083847, A083848, A083849, A002496.
%K A174246 nonn
%O A174246 1,3
%A A174246 _Robert Gerbicz_, Mar 13 2010