A215048 Number of primes of the form 1 + b^4 for 1 < b < 10^n.

3, 17, 110, 789, 6395, 52610, 445868, 3857543, 34057327

Offset: 1

Author

Henryk Dabrowski, Aug 01 2012

Keywords

• nonn

Comments

Primes 1 + b^4 are a form of generalized Fermat primes. It is conjectured that a(n) is asymptotic to 0.66974*li(10^n).

Examples

			a(1) = 3 because the only generalized Fermat primes F_2(b) where b<10^1 are the primes: 17, 257, 1297.
		

References

• Daniel Shanks, On Numbers of the Form n^4 + 1, Math. Comput. 15 (1961), 186-189.

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.

Crossrefs

Cf. A214452.

Programs

Mathematica

Table[Length[Select[Range[2,10^n-1]^4 + 1, PrimeQ]], {n, 5}] (* T. D. Noe, Aug 02 2012 *)

PARI

a(n) = sum(b=1, 10^n/2-1, isprime((2*b)^4+1))

Formula

a(n) = A214452(4*n) - 1.