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

1,3

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

Links

Table of n, a(n) for n=1..48.

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

Example

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.

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

Prog

(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

Crossrefs

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

Sequence in context: A055002 A114097 A325855 * A083847 A034142 A008675

Adjacent sequences: A174243 A174244 A174245 * A174247 A174248 A174249

Keyword

nonn

Author

Robert Gerbicz, Mar 13 2010

Status

approved