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A226318 Positive integers n with p_{n+1}-p_n = 2 and p_{n+3}-p_{n+2} = 2, where p_k denotes the k-th prime.

%I A226318 #17 Sep 08 2022 08:46:05
%S A226318 3,5,26,33,41,43,81,140,142,171,176,234,286,294,313,318,428,458,473,
%T A226318 475,484,577,579,584,671,743,772,862,870,872,891,934,957,1030,1115,
%U A226318 1165,1167,1169,1230,1339,1351,1404,1462,1548,1621,1651,1707,1823,1833,1867,1923,2021,2052,2066,2068,2121,2151,2199,2309,2362
%N A226318 Positive integers n with p_{n+1}-p_n = 2 and p_{n+3}-p_{n+2} = 2, where p_k denotes the k-th prime.
%C A226318 An old conjecture of de Polignac asserts that for any positive even integer d there are infinitely many n>0 with p_{n+1}-p_n = d.
%C A226318 The author has formulated the following further extension.
%C A226318 Conjecture: For any positive even integers d_1,...,d_k, there are infinitely many positive integers n such that p_{n+2j-1}-p_{n+2j-2} = d_j for all j=1,...,k.
%C A226318 For example,
%C A226318 p_{35209566+2j-1}-p_{35209566+2j-2} = 2 for all j = 1,...,7,
%C A226318 p_{19726689+2j-1}-p_{19726689+2j-2} = 6 for all j = 1,...,8,
%C A226318 and p_{297746+2j-1}-p_{297746+2j-2} = 2j for j = 1,2,3,4,5.
%H A226318 Zhi-Wei Sun, <a href="/A226318/b226318.txt">Table of n, a(n) for n = 1..10000</a>
%H A226318 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), 2794-2812.
%H A226318 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1304.5988">On primes in arithmetic progressions</a>, preprint, arXiv:1304.5988.
%H A226318 Yitang Zhang, <a href="http://dx.doi.org/10.4007/annals.2014.179.3.7">Bounded gaps between primes</a>, Annals of Mathematics, Pages 1121-1174 from Volume 179 (2014), Issue 3.
%e A226318 a(1) = 3 and a(2) = 5 since {p_3,p_4}={5,7}, {p_5,p_6}={11,13} and {p_7,p_8}={17,19} are twin prime pairs.
%t A226318 n=0
%t A226318 Do[If[Prime[k+1]-Prime[k]==2&&Prime[k+3]-Prime[k+2]==2,n=n+1;
%t A226318 Print[n," ",k]],{k,1,100}]
%t A226318 PrimePi[#]&/@Transpose[Select[Partition[Prime[Range[2500]],4,1],#[[4]]- #[[3]] == #[[2]]-#[[1]]==2&]][[1]] (* _Harvey P. Dale_, Nov 20 2013 *)
%o A226318 (Magma) [n: n in [1..2500] | (NthPrime(n+1)-NthPrime(n)) eq 2 and (NthPrime(n+3)-NthPrime(n+2)) eq 2]; // _Vincenzo Librandi_, Jun 28 2015
%Y A226318 Cf. A000040, A001359, A006512, A225889, A226115.
%K A226318 nonn
%O A226318 1,1
%A A226318 _Zhi-Wei Sun_, Jun 03 2013