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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

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

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Last modified September 22 08:23 EDT 2018. Contains 315270 sequences. (Running on oeis4.)