A259492 - OEIS

%I #32 Jul 03 2015 06:18:18

%S 4,48852,6,27330,89814,13080,9570,44592,6762,28560,1560,8580,2958,672,

%T 9816,6300,40050,53580,3354,858,4530,100650,182520,49740,48660,25296,

%U 66990,87120,43680,6840,52122,2970,22770,15888,34704,406350,67890,99630,92490,83064,28614,8580,32070,42,50442,38676,818202,30450,47880,4620

%N Least positive integer k such that prime(k)-k, prime(k)+k, prime(k*n)-k*n, prime(k*n)+k*n, prime(k)+k*n and prime(k*n)+k are all prime.

%C Conjecture: Any positive rational number r can be written as m/n with prime(m)-m, prime(m)+m, prime(n)-n, prime(n)+n, prime(m)+n and m+prime(n) all prime.

%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="/A259492/b259492.txt">Table of n, a(n) for n = 1..500</a>

%H Zhi-Wei Sun, <a href="/A259492/a259492_2.txt">Checking the conjecture for r = a/b with a,b = 1..150</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) = 6 since prime(6)-6 = 7, prime(6)+6 = 19, prime(6*3)-6*3 = 43, prime(6*3)+6*3 = 79, prime(6)+6*3 = 31 and prime(6*3)+6 = 67 are all prime.

%t PQ[k_]:=PrimeQ[Prime[k]-k]&&PrimeQ[Prime[k]+k]

%t QQ[m_,n_]:=PQ[m]&&PQ[n]&&PrimeQ[Prime[m]+n]&&PrimeQ[m+Prime[n]]

%t Do[k=0;Label[bb];k=k+1;If[QQ[k,n*k], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k];Continue,{n,1,50}]

%Y Cf. A000040, A258803, A258836, A259487, A259540.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Jun 28 2015