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A174246
Number of primes of the form x^2 + 1 < 2^n.
1
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
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
KEYWORD
nonn
AUTHOR
Robert Gerbicz, Mar 13 2010
STATUS
approved