|
%I #30 Aug 09 2012 13:25:46
%S 3,17,110,789,6395,52610,445868,3857543,34057327
%N Number of primes of the form 1 + b^4 for 1 < b < 10^n.
%C 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).
%D Daniel Shanks, On Numbers of the Form n^4 + 1, Math. Comput. 15 (1961), 186-189.
%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>.
%F a(n) = A214452(4*n) - 1.
%e a(1) = 3 because the only generalized Fermat primes F_2(b) where b<10^1 are the primes: 17, 257, 1297.
%t Table[Length[Select[Range[2,10^n-1]^4 + 1, PrimeQ]], {n, 5}] (* _T. D. Noe_, Aug 02 2012 *)
%o (PARI) a(n) = sum(b=1, 10^n/2-1, isprime((2*b)^4+1))
%Y Cf. A214452.
%K nonn
%O 1,1
%A _Henryk Dabrowski_, Aug 01 2012
| 