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%I #6 Jun 25 2014 12:00:24
%S 754205,1347541,1347542,1347543,1347544,1347545,1692355,2067412,
%T 2067413,2067414,2067415,2218613,2809181,5455635,6127765,6127766,
%U 7034825,7034826,7194143,8603331,8815168,8815169,8815170,8815171,9072188
%N Positive integers n such that all the gaps prime(n+i+1) - prime(n+i) (i = 0, ..., 12) belong to the set {g > 0: g - 1 and g + 1 are twin prime}.
%C Conjecture: For any integer m > 0, there are infinitely many positive integers n such that all those gaps prime(n+i+1) - prime(n+i) (i = 0, ..., m-1) belong to the set {g > 0: g - 1 and g + 1 are twin prime}.
%H Zhi-Wei Sun, <a href="/A244308/b244308.txt">Table of n, a(n) for n = 1..50</a>
%e a(1) = 754205 with those prime(754205+i+1) - prime(754205+i) (i = 0, ..., 12) having respective values 4, 12, 18, 42, 30, 30, 6, 12, 18, 30, 18, 12, 12 in the set {g > 0: g - 1 and g + 1 are twin prime}.
%t d[n_]:=Prime[n+1]-Prime[n]
%t m=0;Do[Do[If[PrimeQ[d[n+i]-1]==False||PrimeQ[d[n+i]+1]==False,Goto[aa]],{i,0,12}];m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,9072188}]
%Y Cf. A000040, A014574, A244254, A244264, A244266, A244294.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 25 2014
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