cp's OEIS Frontend

A261528 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m)+2 is prime with prime(prime(m)+2) = prime(prime(m))+6}.

%I A261528 #17 Aug 23 2015 20:11:19
%S A261528 2,891,81002,814812,86050,5917,65527,109853,2563344,25379,2640232,
%T A261528 266076,775889,67387,68111,37950,353416,347139,56390,11299,89491,
%U A261528 545458,910786,353416,1913477,9025,111569,511796,1456228,37909,1494675,212092,69352,107769,300657,1155675,391972,1073031,55074,49892
%N A261528 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m)+2 is prime with prime(prime(m)+2) = prime(prime(m))+6}.
%C A261528 Conjecture: Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2 is prime with prime(prime(k)+2) = prime(prime(k))+6}.
%C A261528 This implies that there are infinitely many twin prime pairs {p, p+2} with prime(p+2) - prime(p) = 6.
%C A261528 Note that if prime(n+2)-prime(n) = 6 then prime(n+1)-prime(n) = 2 or 4.
%D A261528 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 A261528 Zhi-Wei Sun, <a href="/A261528/b261528.txt">Table of n, a(n) for n = 1..100</a>
%H A261528 Zhi-Wei Sun, <a href="/A261528/a261528.txt">Checking the conjecture for r = a/b with a,b = 1..20</a>
%H A261528 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 A261528 a(1) = 2 since 2*1 = 2, and prime(2)+2 = 3+2 = 5 is prime with prime(5)-prime(3) = 11-5 = 6.
%e A261528 a(2) = 891 since prime(891)+2 = 6947 + 2 = 6949 is prime with prime(6949)-prime(6947) = 70123-70117 = 6, and prime(891*2)+2 = 15269 + 2 = 15271 is prime with prime(15271)-prime(15269) = 167119-167113 = 6.
%t A261528 f[n_]:=Prime[n]
%t A261528 PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
%t A261528 Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,40}]
%Y A261528 Cf. A000040, A001359, A006512, A023201, A046117, A259487, A259540, A261533.
%K A261528 nonn
%O A261528 1,1
%A A261528 _Zhi-Wei Sun_, Aug 23 2015