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%I #28 Sep 07 2022 12:58:21
%S 1,10,2,1,1126,60,55,691,1,24,15,640,5,41,1,671,261,3,8,219,103,1,
%T 1843,128,2240,4,664,12,111,275,19,576,166,5,3,13,7462,243,1,1599,228,
%U 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
%N Least positive integer k such that prime(k) + prime(k*n) is a square.
%C 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.
%C I have verified this conjecture for all those r = a/b with a,b = 1,...,700.
%D 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.
%H Zhi-Wei Sun, <a href="/A259712/b259712.txt">Table of n, a(n) for n = 1..1000</a>
%H Zhi-Wei Sun, <a href="/A259712/a259712_1.txt">Checking the conjecture for r = a/b with a,b = 1..200</a>, 2015.
%H Zhi-Wei Sun, <a href="/A259712/a259712_3.txt">Checking the conjecture for r = a/b with 200 < a,b <= 700 and gcd(a,b) = 1</a>, 2021.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/176r.pdf">Conjectures on representations involving primes</a>, 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 <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e a(2) = 10 since prime(10) + prime(10*2) = 29 + 71 = 10^2.
%e a(5) = 1126 since prime(1126) + prime(1126*5) = 9059 + 55457 = = 64516 = 254^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t 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}]
%t lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k]+Prime[k*n]]],k++];k]; Array[ lpi,70] (* _Harvey P. Dale_, Sep 07 2022 *)
%Y Cf. A000040, A000290, A257856, A259531.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jul 03 2015
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