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

0, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 24, 33, 42, 54, 70, 91, 114, 158, 212, 293, 393, 539, 713, 957, 1301, 1792, 2459, 3378, 4615, 6233, 8418, 11540, 15867, 21729, 29843, 41169, 56534, 77697, 106787, 147067, 203025, 280340, 387308, 535153, 739671

Offset: 1

Author

Robert Gerbicz, Mar 13 2010

Keywords

• nonn

Comments

Terms from Marek Wolf and Robert Gerbicz (code from Robert, computation done by Marek).

It is conjectured that this sequence is unbounded, but this has never been proved. [Comment corrected by Kellen Myers, Oct 12 2014.]

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

Same as A083847 except for a(1) = 0. - Georg Fischer, Oct 14 2018

Examples

			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.
		

Links

• Chris Caldwell, Prime Conjectures and Open Questions
• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
• Eric W. Weisstein, Landau's Problems

Crossrefs

Cf. A083844, A083845, A083846, A083847, A083848, A083849, A002496.

Programs

Maple

N:= 30: # to get a(1) to a(N).
P:= select(isprime,[2,seq((2*i)^2+1, i = 1 .. floor(sqrt(2^N-1)/2))]):
seq(nops(select(`<`,P,2^n)), n=1..N); # Robert Israel, Oct 13 2014

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