A262707 - OEIS

A262707

Positive integers m such that pi(k^2)*pi(m^2) is a square for some 1 < k < m, where pi(x) denotes the number of primes not exceeding x.

%I #12 Aug 22 2019 11:14:12

%S 5,8,10,14,16,19,23,31,35,39,45,63,65,66,68,71,74,82,87,92,94,115,130,

%T 145,151,162,172,204,250,279,292,304,334,391,413,415,418,449,451,454,

%U 461,499,514,524,552,557,626,664,676,683,691,706,708,724,763,766,846,848,858,866

%N Positive integers m such that pi(k^2)*pi(m^2) is a square for some 1 < k < m, where pi(x) denotes the number of primes not exceeding x.

%C Conjecture: The sequence has infinitely many terms.

%C See also A262700 for a related conjecture.

%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 Chai Wah Wu, <a href="/A262707/b262707.txt">Table of n, a(n) for n = 1..1000</a> (n = 1..200 from Zhi-Wei Sun)

%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(2) = 8 since pi(8^2)*pi(2^2) = 18*2 = 6^2.

%e a(3) = 10 since pi(10^2)*pi(3^2) = 25*4 = 10^2.

%t f[n_]:=PrimePi[n^2]

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t n=0;Do[Do[If[SQ[f[x]*f[y]],n=n+1;Print[n," ",y];Goto[aa]],{x,2,y-1}];Label[aa];Continue,{y,1,870}]

%Y Cf. A000290, A000720, A262408, A262443, A262447, A262462, A262698, A262700.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Sep 27 2015