%I #34 Aug 22 2019 00:43:20
%S 5,19,31,151,691,1181,1489,1511,1601,2579,3037,7297,9661,10993,11699,
%T 20407,25657,33937,65099,96419,102911,133157,251789,411841,417271,
%U 670729,808211,1179907,1671277
%N Primes p such that pi(p^2)*pi(q^2) is a square for some prime q < p, where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: (i) The sequence has infinitely many terms.
%C (ii) The Diophantine equation pi(x^n)*pi(y^n) = z^n with n > 2 and x,y,z > 0 has no solution.
%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="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e a(1) = 5 since pi(5^2)*pi(3^2) = 9*4 = 6^2 with 5 and 3 both prime.
%e a(2) = 19 since pi(19^2)*pi(2^2) = 72*2 = 12^2 with 19 and 2 both prime.
%e a(21) = 102911 since pi(102911^2)*pi(919^2) = pi(10590673921)*pi(844561) = 480670430*67230 = 32315473008900 = 5684670^2 with 102911 and 919 both prime.
%e a(22) = 133157 since pi(133157^2)*pi(19^2) = pi(17730786649)*pi(361) = 786299168*72 = 56613540096 = 237936^2 with 133157 and 19 both prime.
%e a(23) = 251789 since pi(251789^2)*pi(10513^2) = pi(63397700521)*pi(110523169) = 2660789341*6331444 = 16846638708338404 = 129794602^2 with 251789 and 10513 both prime.
%t f[n_]:=PrimePi[Prime[n]^2]
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t n=0;Do[Do[If[SQ[f[k]*f[m]],n=n+1;Print[n, " ", Prime[m]];Goto[aa]],{k,1,m-1}];Label[aa];Continue,{m,2,22200}]
%Y Cf. A000040, A000290, A000720, A262408, A262443, A262447, A262462, A262698, A262707.
%K nonn,more
%O 1,1
%A _Zhi-Wei Sun_, Sep 27 2015
%E a(24)-a(29) from _Chai Wah Wu_, Aug 21 2019