A215047 - OEIS

%I #49 Feb 24 2020 00:51:39

%S 3,18,111,840,6655,54109,456361,3954180,34900212,312357933,2826683629,

%T 25814570671,237542444179,2199894223891

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

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

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

%H Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/primes/results.html">Status of the smallest base values yielding Generalized Fermat primes</a>

%H Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/primes/stat.html">How many prime numbers appear in a sequence ?</a>

%H Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/papers/ccdgfpn.html">A Problem on the Conjecture Concerning the Distribution of Generalized Fermat Prime numbers (a new method for the search for large primes)</a>

%H Mersenne Wiki, <a href="http://mersennewiki.org/index.php/User:Merfighters/listtest">Table of known GF primes b^n+1 where n (exponent) is at least 8192</a>.

%H Daniel Shanks, <a href="https://doi.org/10.1090/S0025-5718-1960-0120203-6">On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n^2 + a</a>, Math. Comp. 14 (1960), 320-332.

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

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

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

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

%Y Cf. A083844, A206709.

%K nonn,more

%O 1,1

%A _Henryk Dabrowski_, Aug 01 2012

%E a(13)-a(14) from _Jinyuan Wang_, Feb 23 2020