|
%I #10 Oct 15 2015 17:32:43
%S 1,1,5,2,1,291,4,31,4,131,66,8,113,4,1770,19,122,27,509,61,138,1484,1,
%T 508,118,1033,48,314,78,1522,4,8,169,341,650,37,3456,1172,221,21,119,
%U 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
%N Least positive integer k such that prime(k*n) - prime(k) is a square.
%C 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.
%C This conjecture is a supplement to the conjecture in A259712.
%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="/A257856/b257856.txt">Table of n, a(n) for n = 1..1250</a>
%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.
%e a(3) = 5 since prime(5*3) - prime(5) = 47 - 11 = 6^2.
%e a(70) = 10413 since prime(10413*70) - prime(10413) = 11039173 - 109537 = 3306^2.
%e a(1133) = 697092 since prime(697092*1133) - prime(697092) = 17813555143 - 10523959 = 133428^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t 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}]
%t 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 *)
%Y Cf. A000040, A000290, A257663, A259712.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Jul 12 2015
|
