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A007508
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Number of twin prime pairs below 10^n.
(Formerly M1855)
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48
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2, 8, 35, 205, 1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665, 1177209242304, 10304195697298, 90948839353159, 808675888577436
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OFFSET
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1,1
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COMMENTS
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"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 so-called "cousin"-primes (p; p+4), "sexy"-primes (p; p+6), etc.). - Sergey Pavlov, Apr 08 2021
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REFERENCES
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P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 202.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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D. A. Goldston, J. Pintz & C. Y. Yildirim, Primes in Tuples, I, arXiv:math/0508185 [math.NT], 2005.
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FORMULA
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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
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EXAMPLE
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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 = 2-2 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
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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Cf. A173081 and A181678 (number of twin Ramanujan prime pairs below 10^n).
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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EXTENSIONS
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Added a(17)-a(18) computed by Tomás Oliveira e Silva and link to his web site. - M. F. Hasler, Dec 18 2008
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STATUS
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approved
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