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 A173081 Number of twin prime pairs < 10^n that contain at least one Ramanujan prime (A104272). 1

%I

%S 0,6,28,167,964,6305,45082,335919,2605867

%N Number of twin prime pairs < 10^n that contain at least one Ramanujan prime (A104272).

%C It appears that this counts the number of Ramanujan primes < 10^n that are the lesser prime in a twin prime pair. Equivalently, this sequence also counts the number of Ramanujan primes p with p+2 also prime less than 10^n.

%C It appears that no upper twin prime is a Ramanujan prime without the corresponding lower twin prime also being a Ramanujan prime.

%C This is proved in Section 4 of "Ramanujan Primes: Bounds, Runs, Twins, and Gaps".

%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.

%t nn=50000; t=Table[0,{nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s<nn, t[[s+1]]=k], {k, Prime[3*nn]}]; t=t+1; cnt=0; i=1; Table[While[t[[i]]<10^n, If[PrimeQ[t[[i]]+2], cnt++]; i++]; cnt, {n,Floor[Log[10,t[[-1]]]]}]

%Y Cf. A178128 (Ramanujan primes p such that p+2 is prime), A007508 (number of twin primes pairs < 10^n), A181678 (number of twin Ramanujan primes pairs < 10^n).

%K nonn,more

%O 1,2

%A _T. D. Noe_, Nov 22 2010

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