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

0, 6, 28, 167, 964, 6305, 45082, 335919, 2605867, 20841010, 170395131

Offset

1,2

Comments

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

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

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

Links

Table of n, a(n) for n=1..11.

J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Mathematica

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]]]]}]

Crossrefs

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).

Sequence in context: A344755 A045722 A047129 * A278015 A338810 A345002

Adjacent sequences: A173078 A173079 A173080 * A173082 A173083 A173084

Keyword

nonn,more

Author

T. D. Noe, Nov 22 2010

Extensions

a(10)-a(11) from Dana Jacobsen, Apr 29 2015

Status

approved