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%I #12 Apr 06 2014 22:30:17
%S 0,0,0,1,1,3,3,2,7,6,8,4,9,4,9,8,1,1,3,5,4,6,3,4,4,6,3,11,8,8,7,7,12,
%T 9,4,8,9,12,8,12,8,7,6,7,7,9,4,8,9,11,5,6,3,11,2,5,14,8,8,11,2,1,11,4,
%U 6,4,5,4,1,9,5,2,10,5,4,9,10,11,6,7
%N a(n) = |{0 < k < n: the number of primes in the interval ((k-1)*n, k*n] and the number of primes in the interval (k*n, (k+1)*n] are both prime}|.
%C Conjecture: (i) a(n) > 0 for all n > 3.
%C (ii) For any integer n > 3, there is a prime p < n such that the number of primes in the interval ((p-1)*n, p*n) is a prime.
%C We have verified part (i) for n up to 150000.
%C See also A238277 and A238281 for related conjectures.
%H Zhi-Wei Sun, <a href="/A238278/b238278.txt">Table of n, a(n) for n = 1..3500</a>
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(17) = 1 since the interval (9*17, 10*17] contains exactly 3 primes with 3 prime, and the interval (10*17, 11*17] contains exactly 3 primes with 3 prime.
%t d[k_,n_]:=PrimePi[k*n]-PrimePi[(k-1)n]
%t a[n_]:=Sum[If[PrimeQ[d[k,n]]&&PrimeQ[d[k+1,n]],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,80}]
%Y Cf. A000040, A237578, A237643, A237705, A238224, A238277, A238281.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Feb 22 2014
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