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 A257856 Least positive integer k such that prime(k*n) - prime(k) is a square. 3
 1, 1, 5, 2, 1, 291, 4, 31, 4, 131, 66, 8, 113, 4, 1770, 19, 122, 27, 509, 61, 138, 1484, 1, 508, 118, 1033, 48, 314, 78, 1522, 4, 8, 169, 341, 650, 37, 3456, 1172, 221, 21, 119, 105, 34, 670, 196, 19, 30, 4, 1, 88, 496, 30, 1460, 90, 12, 1270, 812, 2096, 311, 131, 95, 241, 198, 34, 19, 63, 8, 75, 2, 10413 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Conjecture: a(n) exists for any n > 0. In general, every rational number r > 1 can be written as m/n with m > n > 0 such that prime(m) - prime(n) is a square. This conjecture is a supplement to the conjecture in A259712. 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..1250 Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014. EXAMPLE a(3) = 5 since prime(5*3) - prime(5) = 47 - 11 = 6^2. a(70) = 10413 since prime(10413*70) - prime(10413) = 11039173 - 109537 = 3306^2. a(1133) = 697092 since prime(697092*1133) - prime(697092) = 17813555143 - 10523959 = 133428^2. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]] Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[n*k]-Prime[k]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}] lpi[n_]:=Module[{k=1, sq}, sq=Prime[k*n]-Prime[k]; While[!IntegerQ[ Sqrt[ sq]], k++; sq=Prime[k*n]-Prime[k]]; k]; Array[lpi, 70] (* Harvey P. Dale, Oct 15 2015 *) CROSSREFS Cf. A000040, A000290, A257663, A259712. Sequence in context: A174485 A334343 A285641 * A021661 A324187 A219120 Adjacent sequences:  A257853 A257854 A257855 * A257857 A257858 A257859 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jul 12 2015 STATUS approved

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Last modified September 16 07:19 EDT 2021. Contains 347469 sequences. (Running on oeis4.)