cp's OEIS Frontend

A239328 Number of primes p < n with pi(p*n) - pi((p-1)n) prime, where pi(x) denotes the number of primes not exceeding x.

%I A239328 #12 Mar 16 2014 12:00:11
%S A239328 0,0,0,1,1,2,3,2,4,3,4,4,5,5,3,4,1,2,2,5,2,2,2,5,3,3,5,8,4,7,5,5,6,5,
%T A239328 4,4,4,6,7,5,4,6,5,8,7,4,1,2,3,8,5,3,8,5,7,6,6,4,6,9,3,6,7,8,6,3,4,4,
%U A239328 5,6,7,6,11,10,4,6,6,9,6,5
%N A239328 Number of primes p < n with pi(p*n) - pi((p-1)n) prime, where pi(x) denotes the number of primes not exceeding x.
%C A239328 Conjecture: a(n) > 0 for all n > 3. Also, for any integer n > 4, there is a prime p < n with pi((p+1)*n) - pi(p*n) prime.
%C A239328 We have verified that a(n) > 0 for all n = 4, ..., 7*10^5.
%H A239328 Zhi-Wei Sun, <a href="/A239328/b239328.txt">Table of n, a(n) for n = 1..5000</a>
%H A239328 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A239328 a(4) = 1 since 2 and pi(2*4) - pi(1*4) = 4 - 2 = 2 are both prime.
%e A239328 a(5) = 1 since 3 and pi(3*5) - pi(2*5) = 6 - 4 = 2 are both prime.
%e A239328 a(17) = 1 since 11 and pi(11*17) - pi(10*17) = 42 - 39 = 3 are both prime.
%e A239328 a(47) = 1 since 37 ad pi(37*47) - pi(36*47) = 270 - 263 = 7 are both prime.
%t A239328 p[n_,k_]:=PrimeQ[PrimePi[Prime[k]*n]-PrimePi[(Prime[k]-1)*n]]
%t A239328 a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
%t A239328 Table[a[n],{n,1,80}]
%Y A239328 Cf. A000040, A000720, A237578, A238277, A238278, A238281, A239330.
%K A239328 nonn
%O A239328 1,6
%A A239328 _Zhi-Wei Sun_, Mar 16 2014