A347279 -id:A347279 - 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

A347278 First member p(m) of the m-th twin prime pair such that d(m) > 0 and d(m-1) < 0, with d(k) = k/Integral_{x=2..p(k)} 1/log(x)^2 dx - C, C = 2*A005597 = A114907.

Original entry on oeis.org

1369391, 1371989, 1378217, 1393937, 1418117, 1426127, 1428767, 1429367, 1430291, 1494509, 1502141, 1502717, 1506611, 1510307, 35278697, 35287001, 35447171, 35468429, 35468861, 35470271, 35595869, 45274121, 45276227, 45304157, 45306827, 45324569, 45336461, 45336917

Offset: 1

Hugo Pfoertner, Aug 26 2021

nonn

The sequence gives the positions, expressed by A001359(m), where the number of twin prime pairs m seen so far first exceeds the number predicted by the first Hardy-Littlewood conjecture after having been less than the predicted number before. A347279 gives the transitions in the opposite direction.

The total number of twin prime pairs up to that with first member x in the intervals a(k) <= x < A347279(k) is above the Hardy-Littlewood prediction. The total number of twin prime pairs up to that with first member x in the intervals A347279(k) <= x < a(k+1) is below the H-L prediction.

Hugo Pfoertner, Table of n, a(n) for n = 1..12135
Wikipedia, Twin prime, First Hardy-Littlewood conjecture.
Marek Wolf, The Skewes number for twin primes: counting sign changes of pi_2(x)-C_2 Li_2(x), arXiv:1107.2809 [math.NT], 14 Jul 2011.

Cf. A001359, A005597, A114907, A152051, A347279.

a(1) = A210439(2) (Skewes number for twin primes).

halicon(h) = {my(w=Set(vecsort(h)), n=#w, wmin=vecmin(w), distres(v,p)=#Set(v%p)); for(k=1,n, w[k]=w[k]-wmin); my(plim=nextprime(vecmax(w))); prodeuler(p=2, plim, (1-distres(w,p)/p)/(1-1/p)^n) * prodeulerrat((1-n/p)/(1-1/p)^n, 1, nextprime(plim+1))}; \\ k-tuple constant
Li(x, n)=intnum(t=2, n, 1/log(t)^x); \\ logarithmic integral
a347278(nterms,CHL)={my(n=1,pprev=1,np=0); forprime(p=5,, if(p%6!=1&&ispseudoprime(p+2), n++; L=Li(2,p); my(x=n/L-CHL); if(x*pprev>0, if(pprev>0,print1(p,", ");np++; if(np>nterms,return)); pprev=-pprev)))};
a347278(10,halicon([0,2])) \\ computing 30 terms takes about 5 minutes

cp's OEIS Frontend

A007508 Number of twin prime pairs below 10^n.

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