

A007508


Number of twin prime pairs below 10^n.
(Formerly M1855)


48



2, 8, 35, 205, 1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665, 1177209242304, 10304195697298, 90948839353159, 808675888577436
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OFFSET

1,1


COMMENTS

"At the present time (2001), Thomas Nicely has reached pi_2(3*10^15) and his value is confirmed by Pascal Sebah who made a new computation from scratch and up to pi_2(5*10^15) [ = 5357875276068] with an independent implementation."
Though the first paper contributed by D. A. Goldston was reported to be flawed, the more recent one (with other coauthors) maintains and substantiates the result.  Lekraj Beedassy, Aug 19 2005
Theorem. While g is even, g > 0, number of primes p < x (x is integer) such that p' = p + g is also prime, could be written as qpg(x) = qcc(x)  (x  pi(x)  pi(x + g) + 1) where qcc(x) is the number of "common composite numbers" c <= x such that c and c' = c + g both are composite (see Example below; I propose it here as a theorem only not to repeat for socalled "cousin"primes (p; p+4), "sexy"primes (p; p+6), etc.).  Sergey Pavlov, Apr 08 2021


REFERENCES

P. Ribenboim, The Book of Prime Number Records. SpringerVerlag, NY, 2nd ed., 1989, p. 202.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

D. A. Goldston, J. Pintz & C. Y. Yildirim, Primes in Tuples, I, arXiv:math/0508185 [math.NT], 2005.


FORMULA

For 1 < n < 19, a(n) ~ e * pi(10^n) / (5*n  5) = e * A006880(n) / (5*n  5) where e is Napier's constant, see A001113 (probably, so is for any n > 18; we use n > 1 to avoid division by zero).  Sergey Pavlov, Apr 07 2021
For any n, a(n) = qcc(x)  (10^n  pi(10^n)  pi(10^n + 2) + 1) where qcc(x) is the number of "common composite numbers" c <= 10^n such that c and c' = c + 2 both are composite (trivial).  Sergey Pavlov, Apr 08 2021


EXAMPLE

For x = 10, qcc(x) = 4 (since 2 is prime; 4, 6, 8, 10 are even, and no odd 0 < d < 25 such that both d and d' = d + 2 are composite), pi(10) = 4, pi(10 + 2) = 5, but, while v = 2+2 or v = 22 would be even, we must add 1; hence, a(1) = qcc(10^1)  (10^1  pi(10^1)  pi(10^1 + 2) + 1) = 4  (10  4  5 + 1) = 2 (trivial).  Sergey Pavlov, Apr 08 2021


MATHEMATICA

ile = 2; Do[Do[If[(PrimeQ[2 n  1]) && (PrimeQ[2 n + 1]), ile = ile + 1], {n, 5*10^m, 5*10^(m + 1)}]; Print[{m, ile}], {m, 0, 7}] (* Artur Jasinski, Oct 24 2011 *)


PROG



CROSSREFS

Cf. A173081 and A181678 (number of twin Ramanujan prime pairs below 10^n).


KEYWORD

nonn,nice,hard,more


AUTHOR



EXTENSIONS

Added a(17)a(18) computed by Tomás Oliveira e Silva and link to his web site.  M. F. Hasler, Dec 18 2008


STATUS

approved



