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.

5, 8, 10, 14, 16, 19, 23, 31, 35, 39, 45, 63, 65, 66, 68, 71, 74, 82, 87, 92, 94, 115, 130, 145, 151, 162, 172, 204, 250, 279, 292, 304, 334, 391, 413, 415, 418, 449, 451, 454, 461, 499, 514, 524, 552, 557, 626, 664, 676, 683, 691, 706, 708, 724, 763, 766, 846, 848, 858, 866

Offset: 1

Author

Zhi-Wei Sun, Sep 27 2015

Keywords

• nonn

Comments

Conjecture: The sequence has infinitely many terms.

See also A262700 for a related conjecture.

Examples

			a(2) = 8 since pi(8^2)*pi(2^2) = 18*2 = 6^2.
a(3) = 10 since pi(10^2)*pi(3^2) = 25*4 = 10^2.
		

References

• 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.

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000 (n = 1..200 from Zhi-Wei Sun)
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

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

Programs

Mathematica

f[n_]:=PrimePi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
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}]