A259789 Least integer k > 1 such that pi(k)*pi(k*n) is a square, where pi(.) is the prime-counting function given by A000720.

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, 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, 30, 16, 29, 3419, 134, 4, 251, 7

Offset: 1

Author

Zhi-Wei Sun, Jul 05 2015

Keywords

• nonn

Comments

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.

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.

Examples

			a(1) = 2 since pi(2)*pi(2*1) = 1^2.
a(2) = 27 since pi(27)*pi(27*2) = 9*16 = 12^2.
a(8) = 5 since pi(5)*pi(5*8) = 3*12 = 6^2.
a(9) = 96 since pi(96)*pi(96*9) = 24*150 = 60^2.
a(675) = 1465650 since pi(1465650)*pi(1465650*675) = 111747*50331648 = 5624410669056 = 2371584^2.
a(946) = 1922745 since pi(1922745)*pi(1922745*946) = 143599*89749375 = 12887920500625 = 3589975^2.
		

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

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
• Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..60
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
• Zhi-Wei Sun, A new theorem on the prime-counting function, Ramanujan J. 42(2017), 59-67. (See also arXiv:1409.5685 [math.NT], 2014.)
• Zhi-Wei Sun, Conjectures on representations involving primes, 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 arXiv:1211.1588 [math.NT], 2012-2017.)

Crossrefs

Cf. A000040, A000290, A000720, A259712.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
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}]
liks[n_]:=Module[{k=2},While[!IntegerQ[Sqrt[PrimePi[k]PrimePi[k*n]]],k++];k]; Array[liks,60] (* Harvey P. Dale, Jul 12 2024 *)

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