A083847 a(n) = number of primes of the form x^2 + 1 <= 2^n.

1, 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, 1023655, 1416635, 1960813, 2716922, 3764693, 5218926, 7238715

Offset

1,3

Comments

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has never been proved.

References

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Links

Table of n, a(n) for n=1..55.

C. K. Caldwell, An amazing prime heuristic.

Eric Weisstein's World of Mathematics, Landau's Problems.

Prog

(PARI) a(n) = my(nb = 0); forprime(p=2, 2^n, if (issquare(p-1), nb++); ); nb  \\ Michel Marcus, Jun 14 2013

Crossrefs

Cf. A005574, A002496, A083844 - A083849.

Sequence in context: A114097 A325855 A174246 * A034142 A008675 A027581

Adjacent sequences: A083844 A083845 A083846 * A083848 A083849 A083850

Keyword

nonn

Author

Harry J. Smith, May 05 2003

Extensions

More terms from Alexander D. Healy, Feb 06 2005

Status

approved