cp's OEIS Frontend

A259540 Least positive integer k such that k and k*n are terms of A259539.

%I A259540 #23 Jul 04 2015 23:49:28
%S A259540 60,326940,728700,115020,375258,70920,33150,297990,2340,72870,858,
%T A259540 1416210,284130,78978,91368,9438,5547000,767760,1182918,30468,485208,
%U A259540 60,7908810,916188,21522,823968,87720,390210,3252,72870,7878,1823010,1179990,98010,3462,7878,280590,6870,60,434460
%N A259540 Least positive integer k such that k and k*n are terms of A259539.
%C A259540 Conjecture: Any positive rational number r can be written as m/n with m and n terms of A259539.
%C A259540 For example, 4/5 = 11673840/14592300 with 11673840 and 14592300 terms of A259539.
%D A259540 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 A259540 Zhi-Wei Sun, <a href="/A259540/b259540.txt">Table of n, a(n) for n = 1..2000</a>
%H A259540 Zhi-Wei Sun, <a href="/A259540/a259540_2.txt">Checking the conjecture for r = a/b with a,b = 1..100</a>
%H A259540 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 A259540 a(22) = 60 since 60 and 60*22 = 1320 are terms of A259539. In fact, 60-1 = 59, 60+1 = 61, prime(60)+2 = 283, 1320-1 = 1319, 1320+1 = 1321 and prime(1320)+2 = 10861 are all prime.
%t A259540 PQ[k_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]
%t A259540 QQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]+2]
%t A259540 Do[k=0;Label[bb];k=k+1;If[PQ[k]&&QQ[n*(Prime[k]+1)], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]+1];Continue,{n, 1, 40}]
%Y A259540 Cf. A000040, A014574, A258803, A258836, A259487, A259492, A259539.
%K A259540 nonn
%O A259540 1,1
%A A259540 _Zhi-Wei Sun_, Jun 30 2015