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A173081 Number of twin prime pairs < 10^n that contain at least one Ramanujan prime (A104272). 1
0, 6, 28, 167, 964, 6305, 45082, 335919, 2605867, 20841010, 170395131 (list; graph; refs; listen; history; text; internal format)
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: A309490 A045722 A047129 * A278015 A169723 A052395

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

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Last modified October 20 04:37 EDT 2019. Contains 328247 sequences. (Running on oeis4.)