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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
1, 1, 1, 1, 41, 152, 122, 185, 8, 860, 24, 612, 97094 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=0..12.

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


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.


(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))


Cf. A056993, A123599.

Sequence in context: A141957 A108016 A142630 * A082252 A221811 A105100

Adjacent sequences: A259832 A259833 A259834 * A259836 A259837 A259838




Jeppe Stig Nielsen, Jul 06 2015


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



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Last modified November 26 21:15 EST 2022. Contains 358362 sequences. (Running on oeis4.)