A215047 Number of primes of the form 1 + b^2 for 1 < b < 10^n.

3, 18, 111, 840, 6655, 54109, 456361, 3954180, 34900212, 312357933, 2826683629, 25814570671, 237542444179, 2199894223891

Offset: 1

Author

Henryk Dabrowski, Aug 01 2012

Keywords

• nonn
• more

Comments

Primes 1 + b^2 are a form of generalized Fermat primes.

It is conjectured that a(n) is asymptotic to 0.6864067*li(10^n).

Examples

			a(1) = 3 because the only generalized Fermat primes F_1(b) where b < 10^1 are the primes: 5, 17, 37.
		

Links

• Yves Gallot, Status of the smallest base values yielding Generalized Fermat primes
• Yves Gallot, How many prime numbers appear in a sequence ?
• Yves Gallot, A Problem on the Conjecture Concerning the Distribution of Generalized Fermat Prime numbers (a new method for the search for large primes)
• Mersenne Wiki, Table of known GF primes b^n+1 where n (exponent) is at least 8192.
• Daniel Shanks, On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n^2 + a, Math. Comp. 14 (1960), 320-332.

Crossrefs

Cf. A083844, A206709.

Programs

Mathematica

Table[Length[Select[Range[2,10^n-1]^2 + 1, PrimeQ]], {n, 5}] (* T. D. Noe, Aug 02 2012 *)

PARI

a(n) = sum(b=1,10^n/2-1,isprime((2*b)^2+1))

Formula

a(n) = A083844(2*n) - 1.

Extensions

a(13)-a(14) from Jinyuan Wang, Feb 23 2020