A065914 Number of primes in the interval [ 1/2 * q(n), 3/2 * q(n) - 1 ] where q(n) is prime(n)#, the n-th primorial.

1, 3, 8, 38, 294, 2922, 38949, 604764, 11635147, 287020007, 7721129740, 250811981714

Offset: 1

Author

Frank Ellermann, Dec 07 2001

Keywords

• nonn
• more

Comments

Does lim q(n)/a(n+1) converge?

The Prime Number Theorem implies that the limit is 1. [Charles R Greathouse IV, Dec 08 2011]

Examples

			a(2) = 3 primes in [3,9], 9-3 = 6 = q(2) = 3*2. a(3) = 8 primes in [15,45], 45-15 = 30 = q(3) = 5*6. a(4) = 38 primes in [105,315], 315-105 = 210 = q(4) = 7*30.
		

Crossrefs

q(n) = A002110(n), pi(n) = A000720(n).

Programs

PARI

q(n) = prod(k=1, n, prime(k)); \\ A002110
a(n) = my(nb=q(n)); primepi(3*nb/2-1)-primepi(nb/2-1); \\ Michel Marcus, Aug 04 2021

Python

from _future_ import division
from sympy import primepi, primorial
def A065914(n):
    pm = primorial(n)
    return primepi(3*pm//2-1)-primepi(pm//2-1) # Chai Wah Wu, Apr 28 2018

Formula

a(n) = pi( 3*q(n)/2 -1 ) - pi( q(n)/2 -1 ).

Extensions

Corrected by Jason Earls, Dec 19 2001