A261533 - OEIS

%I #10 Apr 26 2020 11:27:06

%S 3,5,59,2789,5231,6947,8087,11717,15269,16229,17207,17909,18059,18131,

%T 24917,28751,35279,37307,39227,39239,41201,43787,45821,47741,51869,

%U 53087,53609,58439,64577,69857,70919,75707,79631,84869,92381,93479,96179,102197,102929,106187

%N Primes p such that p+2 is prime with prime(p+2)-prime(p)=6.

%C The conjecture in A261528 implies that the current sequence has infinitely many terms.

%C Note that for each n > 2 the difference prime(n+2)-prime(n) is at least 6.

%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="/A261533/b261533.txt">Table of n, a(n) for n = 1..2000</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(1) = 3 since 3 and 3+2 = 5 are twin prime, and prime(5)-prime(3) = 11-5 = 6.

%e a(2) = 5 since 5 and 5+2 = 7 are twin prime, and prime(7)-prime(5) = 17-11 = 6.

%t f[n_]:=Prime[n]

%t PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6

%t n=0;Do[If[PQ[k],n=n+1;Print[n," ",f[k]]],{k,1,10119}]

%t Select[Partition[Prime[Range[11000]],2,1],#[[2]]-#[[1]]==2&&Prime[#[[1]]+ 2]- Prime[#[[1]]]==6&][[All,1]] (* _Harvey P. Dale_, Apr 26 2020 *)

%o (PARI) isok(i)=p=prime(i);isprime(p+2)&&prime(p+2)-prime(p)==6;

%o first(m)=my(v=vector(m));i=1;for(j=1,m,while(!isok(i),i++);v[j]=prime(i);i++);v; \\ _Anders Hellström_, Aug 23 2015

%Y Cf. A000040, A001359, A006512, A023201, A046117, A236458, A259488, A259539, A261528.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Aug 23 2015