cp's OEIS Frontend

A237657 a(n) = |{n < m < 2*n: pi(m) and pi(m^2) are both prime}|, where pi(.) is given by A000720.

%I A237657 #7 Feb 10 2014 22:24:14
%S A237657 0,0,0,1,1,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,4,5,5,4,3,
%T A237657 4,4,4,4,4,4,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6,6,6,5,4,4,4,4,5,5,5,5,5,
%U A237657 4,4
%N A237657 a(n) = |{n < m < 2*n: pi(m) and pi(m^2) are both prime}|, where pi(.) is given by A000720.
%C A237657 Conjecture: (i) a(n) > 0 for all n > 8.
%C A237657 (ii) For any integer n > 1 there is a prime p <= n such that n + pi(p) is prime. Also, for n > 5 there is a prime p with n < p < 2*n such that pi(p) is prime.
%C A237657 (iii) For each n > 20, there is a prime p with n < p < 2*n such that pi(p^2) is prime.
%H A237657 Zhi-Wei Sun, <a href="/A237657/b237657.txt">Table of n, a(n) for n = 1..10000</a>
%e A237657 a(4) = 1 since pi(6) = 3 and pi(6^2) = 11 are both prime.
%e A237657 a(10) = 1 since pi(17) = 7 and pi(17^2) = 61 are both prime.
%e A237657 a(17) = 1 since pi(33) = 11 and pi(33^2) = 181 are both prime.
%t A237657 q[n_]:=PrimeQ[PrimePi[n]]&&PrimeQ[PrimePi[n^2]]
%t A237657 a[n_]:=Sum[If[q[m],1,0],{m,n+1,2n-1}]
%t A237657 Table[a[n],{n,1,70}]
%Y A237657 Cf. A000040, A000720, A038107, A237578, A237643, A237656.
%K A237657 nonn
%O A237657 1,18
%A A237657 _Zhi-Wei Sun_, Feb 10 2014