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 A259712 Least positive integer k such that prime(k) + prime(k*n) is a square. 5
 1, 10, 2, 1, 1126, 60, 55, 691, 1, 24, 15, 640, 5, 41, 1, 671, 261, 3, 8, 219, 103, 1, 1843, 128, 2240, 4, 664, 12, 111, 275, 19, 576, 166, 5, 3, 13, 7462, 243, 1, 1599, 228, 6297, 128, 853, 995, 49, 164, 1, 116, 10, 40, 3971, 1741, 32, 338, 11992, 3, 39, 20, 24, 2, 465, 352, 24, 138, 241, 343, 177, 32, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 prime(m) + prime(n) a square. I have verified this conjecture for all those r = a/b with a,b = 1,...,700. 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..200, 2015. Zhi-Wei Sun, Checking the conjecture for r = a/b with 200 < a,b <= 700 and gcd(a,b) = 1, 2021. Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [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.) EXAMPLE a(2) = 10 since prime(10) + prime(10*2) = 29 + 71 = 10^2. a(5) = 1126 since prime(1126) + prime(1126*5) = 9059 + 55457 = = 64516 = 254^2. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]] Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[k]+Prime[k*n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}] CROSSREFS Cf. A000040, A000290, A257856, A259531. Sequence in context: A010174 A073755 A010173 * A343103 A136712 A138999 Adjacent sequences:  A259709 A259710 A259711 * A259713 A259714 A259715 KEYWORD nonn AUTHOR Zhi-Wei Sun, Jul 03 2015 STATUS approved

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Last modified September 24 21:05 EDT 2021. Contains 347651 sequences. (Running on oeis4.)