A228776 Positions of even terms of A050376.

1, 3, 9, 63, 6605, 203286826, 425656284238504569

Offset: 1

Author

Vladimir Shevelev, Sep 04 2013

Keywords

• nonn
• more

Crossrefs

Cf. A050376, A153450 (pi(2^(2^(n-1)))).

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = a[n - 1] + PrimePi[2^(2^(n - 1))]; Array[a, 6] (* Amiram Eldar, Dec 04 2018 *)

PARI

a(n) = if (n==1, 1, a(n-1) + primepi(2^(2^(n-1)))); \\ Michel Marcus, Dec 04 2018

Python

from sympy import primepi
def A228776(n): return sum(primepi(1<<(1<Chai Wah Wu, Feb 18 2025

Formula

For n>=2, a(n) = a(n-1) + pi(2^(2^(n-1))), where pi(x) is the prime counting function.

For s>1, Product_{n>=1} (1 + A050376(a(n))^(-s)) = 2^s/(2^s-1).

A generalization. Let p be a prime. Let for n>=1 the sequence {a^(p)(n)} be sequence of places of terms of A050376 divisible by p. Then, for n>=2, a^(p)(n) = a^(p)(n-1) + pi(p^(2^(n-1))); for s>1, Product_{n>=1} (1 + A050376(a^(p)(n))^(-s)) = p^s/(p^s-1).

Extensions

a(6) from Peter J. C. Moses

a(7) from Jinyuan Wang, Mar 03 2020