A181678 -id:A181678 - cp's OEIS frontend

A007508 Number of twin prime pairs below 10^n.

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

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

"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 an 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

			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

Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 202.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 195.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Richard P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29 (1975), 43-56.
C. K. Caldwell, An amazing prime heuristic, Table 1.
C. K. Caldwell, The Prime Glossary, Twin prime conjecture
T. H. Chan, A note on Primes in Short Intervals, arXiv:math/0503441 [math.NT], 2005.
P. Erdős, Some Unsolved Problems, Michigan Math. J., Volume 4, Issue 3 (1957), 291-300.
G. H. Gadiyar & R. Padma, Renormalisation and the density of prime pairs, arXiv:hep-th/9806061, 1998.
G. H. Gadiyar & R. Padma, Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs, Physica A 269 (1999) 503-510.
D. A. Goldston, J. Pintz & C. Y. Yildirim, Primes in Tuples, I, arXiv:math/0508185 [math.NT], 2005.
D. A. Goldston, J. Pintz & C. Y. Yildirim, Small Gaps Between Primes, II
D. A. Goldston, J. Pintz & C. Y. Yildirim, The Path to Recent Progress on Small Gaps Between Primes, arXiv:math/0512436 [math.NT], 2005-2006.
D. A. Goldston & C. Y. Yildirim, Small Gaps Between Primes, I, arXiv:math/0504336 [math.NT], 2005.
D. A. Goldston & C. Yildirim, Small gaps between consecutive primes
D. A. Goldston et al., Small gaps between primes or almost primes, arXiv:math/0506067 [math.NT], 2005.
D. A. Goldston et al., Small Gaps between Primes Exist, arXiv:math/0505300 [math.NT], 2005.
Xavier Gourdon and Pascal Sebah, Introduction to Twin Primes and Brun's Constant
A. Granville & K. Soundararajan, On the error in Goldston and Yildirim's "Small gaps between consecutive primes"
Gareth A. Jones and Alexander K. Zvonkin, Klein's ten planar dessins of degree 11, and beyond, arXiv:2104.12015 [math.GR], 2021. See p. 24.
P. F. Kelly & F. Pilling, Characterization of the Distribution of Twin Primes, arXiv:math/0103191 [math.NT], 2001.
P. F. Kelly & T. Pilling, Implications of a New Characterization of the Distribution of Twin Primes, arXiv:math/0104205 [math.NT], 2001.
P. F. Kelly & T. Pilling, Discrete Reanalysis of a New Model of the Distribution of Twin Primes, arXiv:math/0106223 [math.NT], 2001.
James Maynard, On the Twin Prime Conjecture, arXiv:1910.14674 [math.NT], 2019, p. 2.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant, Virginia Journal of Science, 46:3 (Fall, 1995), 195-204.
Thomas R. Nicely, Enumeration to 10^14 of the twin primes and Brun's constant [Local copy, pdf only]
Nova Science, Twin Prime Conjecture
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x) [From _M. F. Hasler_, Dec 18 2008]
J. Richstein, Computing the number of twin primes up to 10^14
J. Richstein, Computing the number of twin primes up to 10^14
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.
Jonathan P. Sorenson, Jonathan Webster, Two Algorithms to Find Primes in Patterns, arXiv:1807.08777 [math.NT], 2018.
K. Soundararajan, The distribution of prime numbers, arXiv:math/0606408 [math.NT], 2006.
K. Soundararajan, Small gaps between prime numbers:The work of Goldston-Pintz-Yildirim, arXiv:math/0605696 [math.NT], 2006.
K. Soundararajan, Small gaps between prime numbers:The work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc. 44 (2007), 1-18.
Eric Weisstein's World of Mathematics, Twin Primes
Eric Weisstein, Mathworld Headline News, Twin Primes Proof Proffered
M. Wolf, Some Remarks on the Distribution of twin Primes, arXiv:math/0105211 [math.NT], 2001.
C. Yildirim & D. Goldston, Small gaps between consecutive primes
Index entries for sequences related to numbers of primes in various ranges

Cf. A001097.
Cf. A173081 and A181678 (number of twin Ramanujan prime pairs below 10^n).
Cf. A152051, A347278, A347279.

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 *)

a(n)=my(s,p=2);forprime(q=3,10^n,if(q-p==2,s++);p=q);s \\ Charles R Greathouse IV, Mar 21 2013

Partial sums of A070076(n). - Lekraj Beedassy, Jun 11 2004
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

pi2(10^15) due to Nicely and Szymanski, contributed by Eric W. Weisstein
pi2(10^16) due to Pascal Sebah, contributed by Robert G. Wilson v, Aug 22 2002
Added a(17)-a(18) computed by Tomás Oliveira e Silva and link to his web site. - M. F. Hasler, Dec 18 2008
Definition corrected by Max Alekseyev, Oct 25 2010
a(16) corrected by Dana Jacobsen, Mar 28 2014

A178127 Lesser of twin Ramanujan primes.

Original entry on oeis.org

149, 179, 227, 239, 347, 431, 569, 599, 641, 821, 1019, 1049, 1061, 1427, 1487, 1607, 1787, 1997, 2081, 2129, 2237, 2267, 2657, 2687, 2711, 2789, 2999, 3167, 3257, 3299, 3359, 3527, 3539, 3581, 3671, 3917, 4091, 4127, 4229, 4241, 4337, 4547, 4637, 4649

Offset: 1

Jonathan Sondow, May 20 2010

nonn

By definition, a number p is a member if p and p+2 are Ramanujan primes A104272.
Conjecture: For all n > 570, more than 1/4 of the twin prime pairs < n are both Ramanujan primes.
Motivation for the conjecture is in "Ramanujan primes and Bertrand's postulate" Section 7.
Subsequence of A178128.
See A001359 and A104272 for additional comments, links, and references.

			a(1) = 149 because (149,151) is the 1st pair of twin primes both of which are Ramanujan primes.
11 is not a member even though 11 and 13 are twin primes and 11 is a Ramanujan prime, because 13 is not also a Ramanujan prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..674 from R. J. Mathar)
B. Ghusayni, Subsets of prime numbers, Int. J. Math. Comp. Sci. 7 (2) 2012
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635.
Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A001359, A104272, A164371, A178128, A190654.

Cf. A181678 (number of twin Ramanujan prime pairs less than 10^n).

n := 1:
for i from 1 do
    if A104272(i+1) = A104272(i)+2 then
        printf("%d %d\n",n,A104272(i)) ;
        n := n+1 ;
    end if;
end do: # produces b-file, R. J. Mathar, Sep 21 2017

nn = 1000; R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3*nn]}];
A104272 = R + 1;
twins1 = Position[A104272 // Differences, 2] // Flatten;
A104272[[twins1]] (* Jean-François Alcover, Oct 28 2018, after T. D. Noe in A104272 *)

use ntheory ":all"; my $r = ramanujan_primes(1e5); my @rt = @$r[grep { $r->[$+1]-$r->[$]==2 } 0..$#$r-1]; say "@rt"; # Dana Jacobsen, Sep 06 2015

{A104272(n): A104272(n+1) = A104272(n)+2}.

a(n) = A190654(2n-1) = A190654(2n) - 2.

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A007508 Number of twin prime pairs below 10^n.

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A178127 Lesser of twin Ramanujan primes.

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

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