A062270 -id:A062270 - cp's OEIS frontend

A005597 Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).

6, 6, 0, 1, 6, 1, 8, 1, 5, 8, 4, 6, 8, 6, 9, 5, 7, 3, 9, 2, 7, 8, 1, 2, 1, 1, 0, 0, 1, 4, 5, 5, 5, 7, 7, 8, 4, 3, 2, 6, 2, 3, 3, 6, 0, 2, 8, 4, 7, 3, 3, 4, 1, 3, 3, 1, 9, 4, 4, 8, 4, 2, 3, 3, 3, 5, 4, 0, 5, 6, 4, 2, 3, 0, 4, 4, 9, 5, 2, 7, 7, 1, 4, 3, 7, 6, 0, 0, 3, 1, 4, 1, 3, 8, 3, 9, 8, 6, 7, 9, 1, 1, 7, 7, 9

Offset: 0

N. J. A. Sloane

C_2 = Product_{ p prime > 2} (p * (p-2) / (p-1)^2) is the 2-tuple case of the Hardy-Littlewood prime k-tuple constant (part of First H-L Conjecture): C_k = Product_{ p prime > k} (p^(k-1) * (p-k) / (p-1)^k).
Although C_2 is commonly called the twin prime constant, it is actually the prime 2-tuple constant (prime pair constant) which is relevant to prime pairs (p, p+2m), m >= 1.
The Hardy-Littlewood asymptotic conjecture for Pi_2m(n), the number of prime pairs (p, p+2m), m >= 1, with p <= n, claims that Pi_2m(n) ~ C_2(2m) * Li_2(n), where Li_2(n) = Integral_{2, n} (dx/log^2(x)) and C_2(2m) = 2 * C_2 * Product_{p prime > 2, p | m} (p-1)/(p-2), which gives: C_2(2) = 2 * C_2 as the prime pair (p, p+2) constant, C_2(4) = 2 * C_2 as the prime pair (p, p+4) constant, C_2(6) = 2* (2/1) * C_2 as the prime pair (p, p+6) constant, C_2(8) = 2 * C_2 as the prime pair (p, p+8) constant, C_2(10) = 2 * (4/3) * C_2 as the prime pair (p, p+10) constant, C_2(12) = 2 * (2/1) * C_2 as the prime pair (p, p+12) constant, C_2(14) = 2 * (6/5) * C_2 as the prime pair (p, p+14) constant, C_2(16) = 2 * C_2 as the prime pair (p, p+16) constant, ... and, for i >= 1, C_2(2^i) = 2 * C_2 as the prime pair (p, p+2^i) constant.
C_2 also occurs as part of other Hardy-Littlewood conjectures related to prime pairs, e.g., the Hardy-Littlewood conjecture concerning the distribution of the Sophie Germain primes (A156874) on primes p such that 2p+1 is also prime.
Another constant related to the twin primes is Viggo Brun's constant B (sometimes also called the twin primes Viggo Brun's constant B_2) A065421, where B_2 = Sum (1/p + 1/q) as (p,q) runs through the twin primes.
Reciprocal of the Selberg-Delange constant A167864. See A167864 for additional comments and references. - Jonathan Sondow, Nov 18 2009
C_2 = Product_{prime p>2} (p-2)p/(p-1)^2 is an analog for primes of Wallis' product 2/Pi = Product_{n=1 to oo} (2n-1)(2n+1)/(2n)^2. - Jonathan Sondow, Nov 18 2009
One can compute a cubic variant, product_{primes >2} (1-1/(p-1)^3) = 0.855392... = (2/3) * 0.6601618...* 1.943596... by multiplying this constant with 2/3 and A082695. - R. J. Mathar, Apr 03 2011
Cohen (1998, p. 7) referred to this number as the "twin prime and Goldbach constant" and noted that, conjecturally, the number of twin prime pairs (p,p+2) with p <= X tends to 2*C_2*X/log(X)^2 as X tends to infinity. - Artur Jasinski, Feb 01 2021

			0.6601618158468695739278121100145557784326233602847334133194484233354056423...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 11.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 84-93, 133.
R. K. Guy, Unsolved Problems in Number Theory, Section A8.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 194, 263-264.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..1001
Folkmar Bornemann, PRIMES Is in P: Breakthrough for "Everyman", Notices Amer. Math. Soc., Vol. 50, No. 5 (May 2003), p. 549.
Paul S. Bruckman, Problem H-576, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 39, No. 4 (2001), p. 379; General IZE, Solution to Problem H-576 by the proposer, ibid., Vol. 40, No. 4 (2002), pp. 383-384.
C. K. Caldwell, The Prime Glossary, twin prime constant.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Steven R. Finch, Mathematical Constants, Errata and Addenda, arXiv:2001.00578 [math.HO], 2020, Sec. 2.1.
Philippe Flajolet and Ilan Vardi, Zeta Function Expansions of Classical Constants, Unpublished manuscript. 1996.
Daniel A. Goldston, Timothy Ngotiaoco and Julian Ziegler Hunts, The tail of the singular series for the prime pair and Goldbach problems, Functiones et Approximatio Commentarii Mathematici, Vol. 56, No. 1 (2017), pp. 117-141; arXiv preprint, arXiv:1409.2151 [math.NT], 2014.
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant T_1^(2).
B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
G. Niklasch, Twin primes constant.
Simon Plouffe, The twin primes constant.
Simon Plouffe, Plouffe's Inverter, The twin primes constant.
Pascal Sebah, Numbers, constants and computation (gives 5000 digits).
Eric Weisstein's World of Mathematics, Twin Primes Constant.
Eric Weisstein's World of Mathematics, Twin Prime Conjecture.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.
Eric Weisstein's World of Mathematics, Prime Constellation.
John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.

Cf. A065645 (continued fraction), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271, A114907, A065418 (C_3), A167864, A000010, A008683.

s[n_] := (1/n)*N[ Sum[ MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[ (Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, 160}]; RealDigits[C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 15 2012, after PARI *)
digits = 105; f[n_] := -2*(2^n-1)/(n+1); C2 = Exp[NSum[f[n]*(PrimeZetaP[n+1] - 1/2^(n+1)), {n, 1, Infinity}, NSumTerms -> 5 digits, WorkingPrecision -> 5 digits]]; RealDigits[C2, 10, digits][[1]] (* Jean-François Alcover, Apr 16 2016, updated Apr 24 2018 *)

\p1000; 175/256*prod(k=2,500,(zeta(k)*(1-1/2^k)*(1-1/3^k)*(1-1/5^k)*(1-1/7^k))^(-sumdiv(k,d,moebius(d)*2^(k/d))/k))

prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 12 2021

Equals Product_{k>=2} (zeta(k)*(1-1/2^k))^(-Sum_{d|k} mu(d)*2^(k/d)/k). - Benoit Cloitre, Aug 06 2003
Equals 1/A167864. - Jonathan Sondow, Nov 18 2009
Equals Sum_{k>=1} mu(2*k-1)/phi(2*k-1)^2, where mu is the Möbius function (A008683) and phi is the Euler totient function (A000010) (Bruckman, 2001). - Amiram Eldar, Jan 14 2022

More terms from Vladeta Jovovic, Nov 08 2001
Commented and edited by Daniel Forgues, Jul 28 2009, Aug 04 2009, Aug 12 2009
PARI code removed by D. S. McNeil, Dec 26 2010

A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

Original entry on oeis.org

1, 5, 1, 4, 7, 8, 0, 1, 2, 8, 1, 3, 7, 4, 9, 1, 2, 5, 7, 7, 9, 0, 9, 1, 9, 2, 5, 5, 6, 4, 9, 4, 7, 4, 8, 9, 2, 4, 1, 5, 2, 7, 0, 1, 5, 8, 2, 8, 6, 2, 1, 4, 3, 9, 5, 3, 5, 7, 4, 8, 4, 2, 7, 1, 4, 8, 4, 9, 3, 2, 2, 0, 9, 8, 1, 5, 6, 1, 1, 5, 8, 1, 0, 8, 7, 7, 5, 8, 5, 3, 8, 2, 7, 6, 9, 8, 0, 7, 6, 7, 7, 6, 5, 6, 2

Offset: 1

Jonathan Sondow, Nov 13 2009, Nov 17 2009

Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.
Numerators of partial products are A062271. Denominators are A062270.
An analog for primes of Wallis' product pi/2 = Product_{n >=1} (2n)^2/(2n-1)(2n+1), because A167864 = Product_{prime p>2} (p-1)^2/(p-2)p.
Grosswald (see links) proves that Sum_{k<=x} 2^Omega(k) ~ (1/(8*log(2))) * c * x * (log(x))^2 + O(x * log(x)) where c is this constant. - Amiram Eldar, Jun 06 2020
The asymptotic density of numbers m with A046660(m) = Omega(m) - omega(m) = k is asymptotically ~ c/2^(k+2) as k -> oo, where c is this constant (Rényi, 1955). - Amiram Eldar, Aug 08 2020
Named after the Norwegian mathematician Atle Selberg (1917-2007) and the French mathematician Hubert Delange (1914-2003). - Amiram Eldar, Jun 20 2021

			Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.5147801281374912577909192556...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc., Vol. 18, No. 1 (1954), pp. 83-87.
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.

Michel Balazard, Hubert Delange and Jean-Louis Nicolas, Sur le nombre de facteurs premiers des entiers, C. R. Acad. Sci., Paris, Ser. I, Vol. 306 (1988), pp. 511-514. [From Jonathan Sondow, Nov 17 2009]
Hubert Delange, Sur des formules de Atle Selberg, Acta Arith., Vol. 19 (1971), pp. 105-146.
Steven R. Finch, Mathematical Constants, Errata and Addenda, Sec. 2.1.
Emil Grosswald, The average order of an arithmetic function, Duke Mathematical Journal, Vol. 23, No. 1 (1956), pp. 41-44.
Jean-Louis Nicolas, Sur la distribution des nombres entiers ayant une quantite fixée de facteurs premiers, Acta Arith., Vol. 44 (1984), pp. 191-200.
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.

Cf. A005597.

Cf. A001222 (Omega), A046660, A061142 (2^Omega), A069205 (partial sums of 2^Omega).

s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, 160}]; RealDigits[1/C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 30 2012, after Pari program in A005597 *)
$MaxExtraPrecision = 300; digits = 105; terms = 600; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{3, -2}, {2, 6}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n-1]/(n-1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 19 2016 *)

prodeulerrat((1 + 1/(p*(p-2))),,3) \\ Hugo Pfoertner, Aug 08 2020

Equals 1/A005597.

Equals Product_{prime p>2} (p-1)^2/(p-2)p = (2^2/1*3)(4^2/3*5)(6^2/5*7)(10^2/9*11) ....

A062271 Denominators in partial products of the twin prime constant.

Original entry on oeis.org

4, 64, 256, 1024, 16384, 4194304, 452984832, 603979776, 1073741824, 64424509440, 16698832846848, 8906044184985600, 2244323134616371200, 4588393964104581120, 24471434475224432640, 32628579300299243520

Offset: 2

Frank Ellermann, Jun 16 2001

			a(4)= 256= 2*2*4*4*6*6 / gcd( 3*1*5*3*7*5, 2*2*4*4*6*6 ).

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, ch. 22.20

Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]

A062270 (numerators), A005597 (decimal expansion).

a(n)= a(n-1)*(p(n)-1)^2 / gcd( A062270(n), a(n-1)*(p(n)-1)^2 ) for n > 2.

A065645 Continued fraction for twin prime constant.

Original entry on oeis.org

0, 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, 2, 4, 21, 2, 405, 2, 1, 33, 1, 2, 8, 2, 29, 1, 4, 4, 4, 4, 1, 9, 3, 1, 4, 1, 1, 2, 26, 1, 8, 2, 6, 1, 4, 1, 3, 9, 46, 1, 6, 1, 1, 4, 2, 1, 12, 1, 1, 7, 35, 1, 1, 2, 1, 4, 1199, 2, 3, 1, 2, 3, 3, 13, 15, 4, 1, 1, 1, 10, 9, 6, 3, 1, 3, 1

Offset: 0

Vladeta Jovovic, Nov 08 2001

			1/(1+1/(1+1/(1+1/(16+1/(2+1/(2+1/(2+1/(2+1/(1+1/(18+...)))))))))).

Harry J. Smith, Table of n, a(n) for n = 0..981

Cf. A005597 (decimal expansion), A065646 (denominators of convergents to twin prime constant), A065647 (numerators of convergents to twin prime constant), A062270, A062271.

{ default(realprecision,1002); c2=\
0.66016181584686957392781211001455577843262336028473341331944842333\
5405642304495277143760031413839867911779005226693304002965847755123\
3662277471657132139869687410976206302141537354348531315960978036699\
3213525529976719930247459059310108297829155383446929750520591665713\
3653611991532464281301172462306379341060056466676584434063501649322\
7235289680109349664756004788123579627894598424336557493755818548141\
7362867809870596949870384124336338658931196907915004057371781437108\
1810615401233104810577794415613125444598860988997585328984038108718\
0355252617198871121363828087823497223742240971426974417644552252655\
4899482977179097778404375789195659064999456706290782860882839599039\
4287082529070521554595671723599449769037800675978761690802426600295\
7110920996337082725592846721298580011486979418554018246398874939417\
1182852838236599705032872570808798066220106863047430520199239428201\
4311102297265141514194258422242375342296879836738796224286600285358\
098482833679152235700192585875285961205994728621007171131607980572; x=contfrac(c2); for (n=1, 982, write("b065645.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 15 2009

A191996 Numerators of partial products of a Hardy-Littlewood constant.

Original entry on oeis.org

2, 3, 45, 175, 693, 11011, 2807805, 302307005, 402243205, 714186915, 42803602439, 11086133031701, 5908908905896633, 1488200914442251997, 3041106216468949733, 16213234917387714257, 21611220383343195817, 77778782159652161745383, 67745319261057032880228593

Offset: 2

Wolfdieter Lang, Jun 21 2011

The rational partial products are r(n)=a(n)/A191997(n), n>=1.
The limit r(n), n->infinity, approximately 1.3203236 = A114907, is the constant C(f_1,f_2) appearing in the Hardy-Littlewood conjecture (also called Bateman-Horn conjecture) for the integer polynomials f_1=x and f_2=x+2 (relevant for twin primes). See the Conrad reference Example 1, p. 134, also for the original references.
Essentially the same as A062270. - R. J. Mathar, Jun 23 2011

			The rationals r(n) (in lowest terms) are 2, 3/2, 45/32, 175/128, 693/512, 11011/8192,...

Keith Conrad, Hardy-Littlewood constants, pp. 133-154 in: Mathematical properties of sequences and other combinatorial structures, edts. Jong-Seon No et al., Kluwer, Boston/Dordrecht/London, 2003.

Wolfdieter Lang, Rationals and limit.

A191997, A191998/A191999.

a(n) = numerator(r(n)), with the rational r(n):=2*product(1-1/(p(j)-1)^2,j=2..n), with the primes p(j):=A000040(j).

cp's OEIS Frontend

A005597 Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).

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A167864 Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2))).

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A062271 Denominators in partial products of the twin prime constant.

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A065645 Continued fraction for twin prime constant.

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A191996 Numerators of partial products of a Hardy-Littlewood constant.

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A065646 Denominators of convergents to twin prime constant.

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A065647 Numerators of convergents to twin prime constant.

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