%I
%S 0,1,3,0,3,1,3,3,3,1,5,2,6,3,4,2,6,3,7,3,2,6,8,1,10,3,5,8,9,2,9,6,3,5,
%T 14,5,11,6,9,3,13,8,11,8,8,6,8,8,11,9,6,12,15,10,11,5,11,12,13,9,12,9,
%U 5,17,15,9,18,13,11,12
%N a(n) = {1 <= k <= n: n + pi(k^2) is prime}, where pi(.) is given by A000720.
%C Conjecture: (i) a(n) > 0 for all n > 4.
%C (ii) If n > 1 then n + pi(k*(k1)) is prime for some k = 1, ..., n.
%C (iii) For any integer n > 1, there is a positive integer k <= (n+1)/2 such that pi(n + k*(k+1)/2) is prime.
%C (iv) Any integer n > 1 can be written as p + pi(k*(k+1)/2), where p is a prime and k is among 1, ..., n1.
%H ZhiWei Sun, <a href="/A237595/b237595.txt">Table of n, a(n) for n = 1..3000</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(2) = 1 since 2 + pi(1^2) = 2 is prime.
%e a(6) = 1 since 6 + pi(6^2) = 6 + 11 = 17 is prime.
%e a(10) = 1 since 10 + pi(5^2) = 10 + 9 = 19 is prime.
%e a(21) = 2 since 21 + pi(2^2) = 23 and 21 + pi(9^2) = 43 are both prime.
%e a(24) = 1 since 24 + pi(21^2) = 24 + 85 = 109 is prime.
%t a[n_]:=Sum[If[PrimeQ[n+PrimePi[k^2]],1,0],{k,1,n}]
%t Table[a[n],{n,1,70}]
%Y Cf. A000040, A000720, A237453, A237496, A237497, A237578, A237582.
%K nonn
%O 1,3
%A _ZhiWei Sun_, Feb 09 2014
