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A239884 Least positive integer k <= n with pi(pi(k*n)) a square, or 0 if such a number k does not exist, where pi(x) denotes the number of primes not exceeding x.

%I A239884 #7 Mar 28 2014 22:41:05
%S A239884 1,1,1,1,4,3,3,3,2,2,2,2,2,2,2,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,
%T A239884 3,3,7,7,7,13,6,2,2,2,2,2,2,2,2,2,2,2,2,2,5,16,9,9,9,9,4,4,4,4,4,4,4,
%U A239884 4,4,13,70,20,7,7,7,7,7,7,63,11
%N A239884 Least positive integer k <= n with pi(pi(k*n)) a square, or 0 if such a number k does not exist, where pi(x) denotes the number of primes not exceeding x.
%C A239884 According to part (i) of the conjecture in A238902, a(n) should be always positive. We have verified this for all n = 1, ..., 2*10^5.
%H A239884 Zhi-Wei Sun, <a href="/A239884/b239884.txt">Table of n, a(n) for n = 1..10000</a>
%H A239884 Zhi-Wei Sun, <a href="/A239884/a239884.txt">List of (n, a(n), sqrt(pi(pi(a(n)*n))) for n = 1, ..., 2*10^5</a>
%H A239884 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.
%e A239884 a(5) = 4 since pi(pi(4*5)) = pi(8) = 2^2, but none of pi(pi(1*5)) = pi(3) = 2, pi(pi(2*5)) = pi(4) = 2 and pi(pi(3*5)) = pi(6) = 3 is a square.
%e A239884 a(192969) = 83187 with pi(pi(83187*192969)) = pi(715034817) = 6082^2.
%t A239884 SQ[n_]:=IntegerQ[Sqrt[n]]
%t A239884 f[n_]:=PrimePi[PrimePi[n]]
%t A239884 Do[Do[If[SQ[f[k*n]],Print[n," ",k];Goto[aa]],{k,1,n}];
%t A239884 Print[n," ",0];Label[aa];Continue,{n,1,80}]
%Y A239884 Cf. A000040, A000290, A000720, A238902.
%K A239884 nonn
%O A239884 1,5
%A A239884 _Zhi-Wei Sun_, Mar 28 2014