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%I #19 Jun 20 2013 03:19:36
%S 2,4,2,2,3,2,6,5,3,5,2,3,2,2,2,2,2,2,2,3,5,2,2,4,3,3,2,2,2,3,6,3,2,4,
%T 3,2,2,2,3,3,2,2,2,3,5,2,2,2,3,2,3,3,6,3,4,9,5,2,5,4,2,3,2,3,3,2,4,3,
%U 2,2,5,3,4,4,4,4,3,2,6,2,7,4,2,6,4,2
%N Let (p(n), p(n)+2) be the n-th twin prime pair. a(n) is the smallest k, such that there is only one prime in the interval (k*p(n), k*(p(n)+2)), or a(n)=0, if there is no such k.
%C Conjecture: a(n)>0 for all n.
%H Zak Seidov, <a href="/A218279/b218279.txt">Table of n, a(n) for n = 1..10000</a>
%H V. Shevelev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes</a>, Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.4
%H J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2
%e The first pair of twin primes is (3,5). For k=1 and 2, we have the intervals (3,5) and (6,10), such that not the first but the second interval contains exactly one prime(7). Thus a(1)=2. For n=2 and k=1 to 4, we have the intervals (5,7),(10,14),(15,21), and (20,28) and only the last interval contains exactly one prime(23). Thus, a(2)=4.
%Y Cf. A218275, A166251, A217561, A217566, A217577, A001359, A014574, A006512, A077800.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Oct 25 2012
%E a(6) corrected and terms beyond a(11) contributed by _Zak Seidov_, Oct 25 2012