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A259789
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Least integer k > 1 such that pi(k)*pi(k*n) is a square, where pi(.) is the prime-counting function given by A000720.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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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.)
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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