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

Author

Harry J. Smith, May 05 2003

Keywords

• nonn

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

• C. K. Caldwell, An amazing prime heuristic.
• Eric Weisstein's World of Mathematics, Landau's Problems.

Crossrefs

Cf. A005574, A002496, A083844 - A083849.

Programs

PARI

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

Extensions

More terms from Alexander D. Healy, Feb 06 2005