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%I #54 Mar 28 2014 22:40:43
%S 1,2,1,1,2,3,2,1,2,4,3,4,3,3,3,2,5,5,4,3,5,4,5,4,5,5,6,4,4,6,4,5,4,6,
%T 4,4,3,4,4,3,4,4,4,4,5,3,4,5,4,3,4,5,5,4,2,2,3,2,3,3,3,1,4,3,4,3,3,3,
%U 5,2,1,2,3,5,3,4,4,2,1,5
%N a(n) = |{0 < k <= n: pi(pi(k*n)) is a square}|, where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: (i) a(n) > 0 for all n > 0.
%C (ii) For every n = 1, 2, 3, ..., there exists a positive integer k <= (n+1)/2 such that pi(pi(k*n)) is a triangular number.
%C We have verified parts (i) and (ii) for n up to 2*10^5 and 10^5 respectively.
%C See A239884 for a sequence related to part (i) of the conjecture.
%H Zhi-Wei Sun, <a href="/A238902/b238902.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei 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 pi(pi(3*8)) = pi(pi(24)) = pi(9) = 2^2.
%e a(434) = 1 since pi(pi(297*434)) = pi(pi(128898)) = pi(12064) = 38^2.
%e a(1042) = 1 since pi(pi(698*1042)) = pi(pi(727316)) = pi(58590) = 77^2.
%e a(9143) = 1 since pi(pi(8514*9143)) = pi(pi(77843502)) = pi(4550901) = 565^2.
%e a(48044) > 0 since pi(pi(18332*48044)) = pi(45075237) = 1650^2.
%e a(52158) > 0 since pi(pi(27976*52158)) = pi(72792062) = 2067^2.
%e a(78563) > 0 since pi(pi(26031*78563)) = pi(100326489) = 2404^2.
%e a(98213) > 0 since pi(pi(37308*98213)) = pi(174740922) = 3123^2.
%e a(141589) > 0 since pi(pi(42375*141589)) = pi(279538049)= 3899^2.
%e a(154473) > 0 since pi(pi(42954*154473)) = pi(307695484) = 4080^2.
%e a(195387) > 0 since pi(pi(60161*195387)) = pi(530982180) = 5282^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t p[k_,n_]:=SQ[PrimePi[PrimePi[k*n]]]
%t a[n_]:=Sum[If[p[k,n],1,0],{k,1,n}]
%t Table[a[n],{n,1,80}]
%o (PARI) {a(n) = sum( k=1, n, issquare( primepi( primepi( k*n))))}; /* _Michael Somos_, Mar 10 2014 */
%Y Cf. A000040, A000217, A000290, A000720, A237598, A237840, A238504, A239884.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Mar 06 2014
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