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

1, 1, 1, 1, 41, 152, 122, 185, 8, 860, 24, 612, 97094

Offset: 0

Author

Jeppe Stig Nielsen, Jul 06 2015

Keywords

• nonn
• more
• hard

Comments

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.

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.

a(12) is expected to be near 97000.

Examples

			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.
		

Links

• Y. Gallot, Status of the smallest base values yielding Generalized Fermat primes.

Crossrefs

Cf. A056993, A123599.

Programs

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))

Extensions

a(12) via b-file of A088362 from Jeppe Stig Nielsen, Feb 16 2022