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%I #15 Apr 06 2014 22:20:03
%S 0,0,1,2,1,1,1,1,2,2,2,4,3,3,3,1,1,2,2,3,3,1,1,2,3,4,4,4,4,6,5,4,4,2,
%T 2,3,3,5,5,3,3,3,3,4,4,3,3,3,3,3,3,2,2,4,5,5,5,4,4,7,6,5,5,4,4,5,5,7,
%U 7,5
%N Number of primes p < n with pi(n-p) a square, where pi(.) is given by A000720.
%C Conjecture: (i) a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 5, 6, 7, 8, 16, 17, 22, 23, 148, 149.
%C (ii) For any integer n > 2, there is a prime p < n with pi(n-p) a triangular number.
%C We have verified that a(n) > 0 for every n = 3, ..., 1.5*10^7. See A237710 for the least prime p < n with pi(n-p) a square.
%C See also A237705, A237720 and A237721 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A237706/b237706.txt">Table of n, a(n) for n = 1..10000</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(8) = 1 since 7 is prime with pi(8-7) = 0^2.
%e a(16) = 1 since 7 is prime with pi(16-7) = 2^2.
%e a(149) = 1 since 139 is prime with pi(149-139) = pi(10) = 2^2.
%e a(637) = 2 since 409 is prime with pi(637-409) = pi(228) = 7^2, and 613 is prime with pi(637-613) = pi(24) = 3^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t q[n_]:=SQ[PrimePi[n]]
%t a[n_]:=Sum[If[q[n-Prime[k]],1,0],{k,1,PrimePi[n-1]}]
%t Table[a[n],{n,1,70}]
%Y Cf. A000040, A000217, A000290, A000720, A237598, A237612, A237705, A237710, A237720, A237721.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Feb 11 2014
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