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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. 2
1, 3, 8, 38, 294, 2922, 38949, 604764, 11635147, 287020007, 7721129740, 250811981714 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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]

LINKS

Table of n, a(n) for n=1..12.

FORMULA

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

EXAMPLE

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.

PROG

(PARI) pi(x)=c=0; forprime(p=2, x, c++); c q(n) = prod(k=1, n, prime(k)) a(n) = pi(3*q(n)/2-1)-pi(q(n)/2-1) for(n=1, 11, print1(a(n), ", "))

(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

CROSSREFS

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

Sequence in context: A264657 A190658 A106558 * A288759 A180368 A108262

Adjacent sequences:  A065911 A065912 A065913 * A065915 A065916 A065917

KEYWORD

nonn,more

AUTHOR

Frank Ellermann, Dec 07 2001

EXTENSIONS

Corrected by Jason Earls, Dec 19 2001

STATUS

approved

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Last modified September 15 20:11 EDT 2019. Contains 327086 sequences. (Running on oeis4.)