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%I #32 Dec 17 2017 10:07:53
%S 2,27,8,2,2,9,3,5,96,10,9,2,2,2,28,4,9,11,8,195,3,3,723,28,573,225,2,
%T 2,2,35,46,132,4,4,65,14,58,11,8,967,311,10,98,3,3,21,94,20,2,2,28,23,
%U 30,16,29,3419,134,4,251,7
%N Least integer k > 1 such that pi(k)*pi(k*n) is a square, where pi(.) is the prime-counting function given by A000720.
%C Conjecture: a(n) exists for any n > 0. In general, every positive rational number r can be written as m/n, where m and n are positive integers with pi(m)*pi(n) a positive square.
%C For example, 25/32 = 13102500/16771200 with pi(13102500)*pi(16771200) = 855432*1077512 = 921738245184 = 960072^2, and 49/58 = 1076068567/1273713814 with pi(1076068567)*pi(1273713814) = 54511776*63975626 = 3487424993971776 = 59054424^2.
%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="/A259789/b259789.txt">Table of n, a(n) for n = 1..1000</a>
%H Zhi-Wei Sun, <a href="/A259789/a259789_2.txt">Checking the conjecture for r = a/b with a,b = 1..60</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.
%H Zhi-Wei Sun, <a href="http://link.springer.com/article/10.1007/s11139-015-9702-z">A new theorem on the prime-counting function</a>, Ramanujan J. 42(2017), 59-67. (See also <a href="http://arxiv.org/abs/1409.5685">arXiv:1409.5685 [math.NT]</a>, 2014.)
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/176r.pdf">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e a(1) = 2 since pi(2)*pi(2*1) = 1^2.
%e a(2) = 27 since pi(27)*pi(27*2) = 9*16 = 12^2.
%e a(8) = 5 since pi(5)*pi(5*8) = 3*12 = 6^2.
%e a(9) = 96 since pi(96)*pi(96*9) = 24*150 = 60^2.
%e a(675) = 1465650 since pi(1465650)*pi(1465650*675) = 111747*50331648 = 5624410669056 = 2371584^2.
%e a(946) = 1922745 since pi(1922745)*pi(1922745*946) = 143599*89749375 = 12887920500625 = 3589975^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t Do[k=1; Label[bb]; k=k+1; If[SQ[PrimePi[k]*PrimePi[k*n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue,{n,1,60}]
%o (PARI) main(size) = {v=vector(size);for(t=1,size,v[t]=1;until(issquare(primepi(v[t])*primepi(t*v[t])),v[t]++));return(v);} \\ _Anders Hellström_, Jul 05 2015
%Y Cf. A000040, A000290, A000720, A259712.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jul 05 2015
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