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A218769 Let (p,p+2) be the n-th twin prime pair. a(n) is the least integer r > 1 for which the interval (r*p, r*(p+2)) contains no primes, or a(n)=0, if no such r exists.

%I A218769 #41 Nov 16 2014 06:46:55
%S A218769 0,0,0,0,4,0,2,2,2,2,3,2,5,5,4,5,4,4,3,2,2,4,4,2,2,2,6,3,3,4,3,2,3,2,
%T A218769 2,7,3,3,2,2,2,6,0,3,2,2,5,5,23,2,3,2,2,2,2,2,2,2,3,2,2,3,2,5,2
%N A218769 Let (p,p+2) be the n-th twin prime pair. a(n) is the least integer r > 1 for which the interval (r*p, r*(p+2)) contains no primes, or a(n)=0, if no such r exists.
%C A218769 For n<=20000, the largest a(n) is a(49)=23. a(n)=0 for n = 1, 2, 3, 4, 6, 43, 37890, 606457, ... corresponding to the twin primes (p, p+2) with p=3, 5, 11, 17, 41, 1277, 5995727, 143556431, ....
%H A218769 Peter J. C. Moses, <a href="/A218769/b218769.txt">Table of n, a(n) for n = 1..20000</a>
%H A218769 Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Moses/moses1.html">On intervals (kn, (k+1)n) containing a prime for all n>1</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. <a href="http://arxiv.org/abs/1212.2785">arXiv:1212.2785</a>
%H A218769 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 A218769 The 13th twin prime pair is {179, 181}. For r = 2 the range {358, ..., 362} contains prime 359; for r = 3, the range {537, ..., 543} contains prime 541; for r = 4, the range {716, ..., 724} contains prime 719. But for r = 5, the range {895, ..., 905} does not contain any prime. Thus a(13) = 5.
%t A218769 rmax = 100; p1[1] = 3; p1[n_] := p1[n] = (p = NextPrime[p1[n-1]]; While[ !PrimeQ[p+2], p = NextPrime[p]]; p); a[n_] := Catch[ For[r = 2, r <= rmax, r++, If[ PrimePi[r*p1[n]] == PrimePi[r*(p1[n] + 2)], Throw[r], If[r == rmax, Throw[0]]]]]; Table[ a[n] , {n, 1, 65}] (* _Jean-François Alcover_, Dec 13 2012 *)
%Y A218769 Cf. A218279, A001359, A218561, A195270, A195271, A195325, A195377, A195379.
%K A218769 nonn
%O A218769 1,5
%A A218769 _Vladimir Shevelev_, _Peter J. C. Moses_, and _Charles R Greathouse IV_, Nov 05 2012
%E A218769 Typo in definition corrected by _Jonathan Sondow_, Dec 21 2012