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%I #24 Aug 05 2021 12:49:25
%S 1,3,8,38,294,2922,38949,604764,11635147,287020007,7721129740,
%T 250811981714
%N 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.
%C Does lim q(n)/a(n+1) converge?
%C The Prime Number Theorem implies that the limit is 1. [_Charles R Greathouse IV_, Dec 08 2011]
%F a(n) = pi( 3*q(n)/2 -1 ) - pi( q(n)/2 -1 ).
%e 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.
%o (Python)
%o from __future__ import division
%o from sympy import primepi, primorial
%o def A065914(n):
%o pm = primorial(n)
%o return primepi(3*pm//2-1)-primepi(pm//2-1) # _Chai Wah Wu_, Apr 28 2018
%o (PARI) q(n) = prod(k=1, n, prime(k)); \\ A002110
%o a(n) = my(nb=q(n)); primepi(3*nb/2-1)-primepi(nb/2-1); \\ _Michel Marcus_, Aug 04 2021
%Y q(n) = A002110(n), pi(n) = A000720(n).
%K nonn,more
%O 1,2
%A _Frank Ellermann_, Dec 07 2001
%E Corrected by _Jason Earls_, Dec 19 2001
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