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%I #14 Apr 06 2014 22:19:12
%S 1,1,1,2,2,2,4,3,5,2,3,5,3,6,1,2,3,3,5,3,5,2,6,4,4,5,3,6,4,3,2,5,3,4,
%T 3,4,4,3,6,4,3,4,2,1,2,9,3,4,4,4,5,7,4,7,3,6,7,3,7,7,5,1,4,5,3,3,10,5,
%U 4,7
%N a(n) = |{0 < k < prime(n): pi(k*n) is a square}|, where pi(.) is given by A000720.
%C Conjecture: (i) a(n) > 0 for all n > 0.
%C (ii) For each n > 9, there is a positive integer k < prime(n)/2 such that pi(k*n) is a triangular number.
%C See also A237612 for the least k > 0 with pi(k*n) a square.
%H Zhi-Wei Sun, <a href="/A237598/b237598.txt">Table of n, a(n) for n = 1..2500</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(3) = 1 since pi(3*3) = 2^2 with 3 < prime(3) = 5.
%e a(6) = 2 since pi(4*6) = 3^2 with 4 < prime(6) = 13, and pi(9*6) = 4^2 with 9 < prime(6) = 13.
%e a(15) = 1 since pi(28*15) = 9^2 with 28 < prime(15) = 47.
%e a(62) = 1 since pi(68*62) = 24^2 with 68 < prime(62) = 293.
%e a(459) = 1 since pi(2544*459) = 301^2 with 2544 < prime(459) = 3253.
%t sq[n_]:=IntegerQ[Sqrt[PrimePi[n]]]
%t a[n_]:=Sum[If[sq[k*n],1,0],{k,1,Prime[n]-1}]
%t Table[a[n],{n,1,70}]
%Y Cf. A000040, A000217, A000290, A000720, A237578, A237597, A237612, A237614.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Feb 10 2014
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