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A259835 a(n) is the number of odd primes of the form b^(2^n)+1 that are less than A123599(n+1). 1

%I

%S 1,1,1,1,41,152,122,185,8,860,24,612

%N a(n) is the number of odd primes of the form b^(2^n)+1 that are less than A123599(n+1).

%C A generalized Fermat prime b^(2^n)+1 can be thought of as belonging to the "family" n. Then a(n) counts how many generalized Fermat primes in family n precede the first generalized Fermat prime in family n+1.

%C Each family as defined here is a subset of its preceding family, in the sense that b^(2^n) + 1 = (b^2)^(2^(n-1)) + 1.

%C a(12) is expected to be near 97000.

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

%e To find a(5), find all primes b^32 + 1 until you reach a base b that is a perfect square. In this case you find 152 nonsquare b values { 30, 54, 96, 112, ..., 10396 }, but the 153rd b is 10404, a perfect square. So 10404^32 + 1 = 102^64 + 1 belongs to the next family. Therefore a(5)=152.

%o (PARI) b=2;for(n=0,100,x=0;until(,if(ispseudoprime(b^(2^n)+1),if(issquare(b,&b),break,x++));b+=2);print("a(",n,")=",x,", next b is ",b))

%Y Cf. A056993, A123599.

%K nonn,more,hard

%O 0,5

%A _Jeppe Stig Nielsen_, Jul 06 2015

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Last modified November 28 04:31 EST 2021. Contains 349400 sequences. (Running on oeis4.)