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%I #38 Nov 16 2017 01:48:11
%S 0,1,1,1,0,1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,1,1,1,1,0,0,1,0,0,2,0,1,0,2,
%T 0,0,1,1,0,1,0,0,0,1,0,2,2,1,0,0,0,0,0,1,1,0,0,1,1,0,0,2,0,0,1,1,1,1,
%U 0,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0,1,1
%N Number of primes between prime(n)/2 and prime(n+1)/2.
%C Conjecture: this sequence is unbounded, as implied by Dickson's conjecture. - _Charles R Greathouse IV_, Oct 09 2012
%C Conjecture: 0 appears infinitely often. - _Jon Perry_, Oct 10 2012
%C First differences of A079952. - _Peter Munn_, Oct 19 2017
%H Hans Havermann, <a href="/A217564/b217564.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = pi(prime(n + 1)/2) - pi(prime(n)/2), where pi is the prime counting function and prime(n) is the n-th prime.
%F Equivalently, a(n) = A079952(n+1) - A079952(n). - _Peter Munn_, Oct 19 2017
%F The average order of a(n) is 1/2, that is, a(1) + a(2) + ... + a(n) ~ n/2. - _Charles R Greathouse IV_, Oct 09 2012
%e a(30) = 2 because there are two primes between prime(30)/2 [=113/2] and prime(31)/2 [=127/2]; i.e., the numbers 59 and 61.
%t q = 2; Table[p = q; q = NextPrime[p]; Length[Position[PrimeQ[Range[p + 1, q - 1, 2]/2], True]], {105}]
%t Table[PrimePi[Prime[n + 1]/2] - PrimePi[Prime[n]/2], {n, 105}] (* _Alonso del Arte_, Oct 08 2012 *)
%Y Cf. A079952, A102820.
%Y Cf. A215237 (location of first n).
%Y A164368 lists the prime(n) corresponding to the zero terms.
%K nonn
%O 1,30
%A _Hans Havermann_, Oct 06 2012
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