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%I #5 Sep 23 2015 03:58:00
%S 8,11,14,19,20,36,38,45,66,87,91,115,139,143,152,155,201,220,227,279,
%T 357,383,391,415,418,452,476,480,489,496,500,514,521,524,549,552,557,
%U 588,595,632,653,676,706,708,749,753,761,766,820,846,863,877,922,1009,1038,1041,1044,1052,1057,1080
%N Positive integers m such that pi(m^2) = pi(j^2)*pi(k^2) for some 0 < j < k < m, where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: (i) The sequence has infinitely many terms. Also, there are infinitely many positive integers m such that pi(m^2) = pi(j^2)*pi(k^2) for no 0 < j <= k < m.
%C (ii) For any integer n > 2, the equation pi(x^n)*pi(y^n) = pi(z^n) has no solution with 0 < x <= y < z.
%D Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H Zhi-Wei Sun, <a href="/A262443/b262443.txt">Table of n, a(n) for n = 1..300</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e a(1) = 8 since pi(8^2) = pi(64) = 18 = 2*9 = pi(2^2)*pi(5^2) with 0 < 2 < 5 < 8.
%e a(4) = 19 since pi(19^2) = pi(361) = 72 = 4*18 = pi(3^2)*pi(8^2) with 0 < 3 < 8 < 19.
%t f[n_]:=PrimePi[n^2]
%t T[n_]:=Table[f[k],{k,1,n}]
%t Dv[n_]:=Divisors[f[n]]
%t Le[n_]:=Length[Dv[n]]
%t n=0;Do[Do[If[MemberQ[T[m],Part[Dv[m],i]]&&MemberQ[T[m],Part[Dv[m],Le[m]-i+1]],n=n+1;Print[n," ",m];Goto[aa]],{i,2,(Le[m]-1)/2}];Label[aa];Continue,{m,1,1080}]
%Y Cf. A000290, A000720, A038107, A262408, A262409.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Sep 23 2015
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