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

1,1

Comments

Conjecture: The sequence has infinitely many terms.

See also A262700 for a related conjecture.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A098594 A154315 A182084 * A257046 A314382 A022799

Adjacent sequences: A262704 A262705 A262706 * A262708 A262709 A262710

Keyword

nonn

Author

Zhi-Wei Sun, Sep 27 2015

Status

approved