A334051 The difference between the number of prime numbers in the ranges (1, p_n] and (p_n, 2*p_n], where p_n is the n-th prime.

0, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 4, 5, 6, 5, 4, 6, 6, 6, 8, 7, 8, 8, 6, 6, 8, 9, 11, 12, 8, 9, 8, 9, 8, 10, 9, 10, 11, 12, 11, 12, 11, 12, 13, 14, 12, 10, 11, 12, 12, 13, 14, 13, 13, 13, 15, 16, 17, 18, 19, 18, 14, 14, 16, 17, 13, 14, 13, 15, 16, 17, 16, 16

Offset: 1

Author

Ya-Ping Lu, Sep 11 2020

Keywords

• nonn

Examples

			a(6) = 2*6 - pi(2*p_6) = 12 - pi(2*13) = 12 - pi(26) = 12 - 9 = 3.
		

Crossrefs

Cf. A000040, A000720, A020900, A331677.

Programs

PARI

a(n) = 2*n - primepi(2*prime(n)); \\ Michel Marcus, Sep 11 2020

Python

from sympy import prime, primepi
for n in range(1, 10001):
    a = 2*n - primepi(2*prime(n))
    print(a)

Formula

a(n) = 2*n - pi(2*p_n).

a(n) = 2*n - A020900(n). - Michel Marcus, Sep 11 2020

a(n) = 1 + A331677(n). - Alois P. Heinz, Oct 09 2020