A152051 -id:A152051 - 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

A160910 Decimal expansion of c = sum over twin primes (p, p+2) of (1/p^2 + 1/(p+2)^2).

Original entry on oeis.org

2, 3, 7, 2, 5, 1, 7, 7, 6, 5, 7

Offset: 0

William Royle (seriesandsequences(AT)yahoo.com), May 29 2009

Compare Viggo Brun's constant (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + (1/29 + 1/31) + ... (see A065421, A005597).
It appears that c = Sum 1/A001359(n)^2 + 1/A006512(n)^2. - R. J. Mathar, May 30 2009
0.237251776574746 < c < 0.237251776947124. - Farideh Firoozbakht, May 31 2009
c < 0.2725177657771. - Hagen von Eitzen, Jun 03 2009
From Farideh Firoozbakht, Jun 01 2009: (Start)
We can show that a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
Proof: s1 = 0.237251776576249072... is the sum up to prime(499,000,000) and s2 = 0.237251776576250009... is the sum up to prime(500,000,000).
By using the fact that number of twin primes between the first 10^6*n primes and the first 10^6*(n+1) primes is decreasing (up to the first 2*10^9 primes), we conclude that the sum up to prime(2,000,000,000) is less than s2 + 1500*(s2-s1).
But since s2-s1 < 10^(-15), the sum up to prime(2*10^9) is less than s2 + 1.5*10^(-12) = 0.237251776576250009... + 1.5*10^(-12) = 0.237251776577550009... .
Hence the constant c is less than
0.237251776577550009... + lim(sum(1/k^2,{k, prime(2,000,000,001), n}, n -> infinity)
< 0.237251776577550009... + 2.12514*10^(-11)
< 0.237251776598801409.
So we have 0.237251776576250009 < c < 0.237251776598801409, hence a(9)=6, a(10)=5, and a(11) is in the set {7, 8, 9}.
I guess that a(11)=7. (End)
From Jon E. Schoenfield, Jan 02 2019: (Start)
Given that the Hardy-Littlewood approximation to the number of twin prime pairs < y is
     2 * C_2 * Integral_{x=2..y} dx/log(x)^2
  where C_2 = 0.660161815846869573927812110014555778432623 (see A152051), we can estimate the size of the tail of the summation Sum(1/A001359(j)^2) + 1/A006512(j)^2) for twin primes > y as
     t(y) = 2 * C_2 * Integral_{x>y} 2*dx/(x*log(x))^2.
Let s(y) be the sum of the squares of the reciprocals of all the twin primes <= y, and let s'(y) = s(y) + t(y) be the result of adding to the actual value s(y) the estimated tail size t(y). Evaluating s(y), t(y), and s'(y) at y = 2^d for d = 20..33 gives
.
   d        s(2^d)        t(2^d)*10^10    s(2^d) + t(2^d)
  == ==================== ============ ====================
  20 0.237251764919808326 115.34589710 0.237251776454398036
  21 0.237251771317612979  52.59702970 0.237251776577315949
  22 0.237251774173347724  24.08221952 0.237251776581569676
  23 0.237251775469086555  11.06766714 0.237251776575853269
  24 0.237251776066813995   5.10395459 0.237251776577209454
  25 0.237251776340760021   2.36119196 0.237251776576879217
  26 0.237251776467109357   1.09553336 0.237251776576662693
  27 0.237251776525743797   0.50967952 0.237251776576711749
  28 0.237251776552887645   0.23771866 0.237251776576659511
  29 0.237251776565549906   0.11113468 0.237251776576663374
  30 0.237251776571456873   0.05207020 0.237251776576663892
  31 0.237251776574218065   0.02444677 0.237251776576662742
  32 0.237251776575513036   0.01149984 0.237251776576663020
  33 0.237251776576121140   0.00541938 0.237251776576663078
.
which agrees with all the terms in the Data section and suggests likely values for additional terms.
(End)

			(1/9 + 1/25) + (1/25 + 1/49) + (1/121 + 1/169) + (1/289 + 1/361) + (1/841 + 1/961) + ... = 0.237251...

Various authors, On the computation of A160910

Cf. A001359, A005597, A006512, A065421, A152051.

R. J. Mathar pointed out that the value of c as originally submitted was incorrect (see link). - N. J. A. Sloane, May 31 2009
More terms from Farideh Firoozbakht and Hagen von Eitzen, Jun 01 2009
Name changed by Michael B. Porter, Jan 04 2019

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A160910 Decimal expansion of c = sum over twin primes (p, p+2) of (1/p^2 + 1/(p+2)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI