A065418 -id:A065418 - 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

A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.

Original entry on oeis.org

1, 3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880, 16104, 44220, 122640, 341484, 956576, 2690010, 7596480, 21522228, 61171656, 174336264, 498111952, 1426403748, 4093181688, 11767874940, 33891544368, 97764009000, 282429535752, 817028131140, 2366564736720, 6863037256208, 19924948267224, 57906879556410

Offset: 0

N. J. A. Sloane

Number of Lyndon words of length n on {1,2,3}. A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts. - John W. Layman, Jan 24 2006
Exponents in an expansion of the Hardy-Littlewood constant Product(1 - (3*p - 1)/(p - 1)^3, p prime >= 5), whose decimal expansion is in A065418: the constant equals Product_{n >= 2} (zeta(n)*(1 - 2^(-n))*(1 - 3^(-n)))^(-a(n)). - Michael Somos, Apr 05 2003
Number of aperiodic necklaces with n beads of 3 colors. - Herbert Kociemba, Nov 25 2016
Number of irreducible harmonic polylogarithms, see page 299 of Gehrmann and Remiddi reference and table 1 of Maître article. - F. Chapoton, Aug 09 2021
For n>=2, a(n) is the number of Hesse loops of length 2*n, see Theorem 22 of Dutta, Halbeisen, Hungerbühler link. - Sayan Dutta, Sep 22 2023
For n>=2, a(n) is the number of orbits of size n of isomorphism classes of elliptic curves under the Hesse derivative, see Theorem 2 of Kettinger link. - Jake Kettinger, Aug 07 2024

			For n = 2 the a(2)=3 polynomials are  x^2+1, x^2+x+2, x^2+2*x+2. - _Robert Israel_, Dec 16 2015

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Seiichi Manyama, Table of n, a(n) for n = 0..2102 (terms 0..200 from T. D. Noe)
Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 4th line of Table 1 (p. 6).
Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 6.
Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, Properties of Hesse derivatives of cubic curves, arXiv:2309.05048 [math.AG], 2023.
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms. Comput. Phys. Comm. 141 (2001), no. 2, 296-312.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. See Table A.2.
Jake Kettinger, The dynamics of the Hesse derivative on the j-invariant, arXiv:2408.04117 [math.AG], 2024.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Computer Physics Communications, Volume 174, Issue 3, 1 February 2006, Pages 222-240.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag, 1978.
Index entries for sequences related to Lyndon words

Column 3 of A074650.

Cf. A000031, A001037, A001693, A001867, A027375, A027377, A054718, A102660.

with(numtheory): A027376 := n -> `if`(n = 0, 1,
add(mobius(d)*3^(n/d), d = divisors(n))/n):
seq(A027376(n), n = 0..32);

a[0]=1; a[n_] := Module[{ds=Divisors[n], i}, Sum[MoebiusMu[ds[[i]]]3^(n/ds[[i]]), {i, 1, Length[ds]}]/n]
a[0]=1; a[n_] := DivisorSum[n, MoebiusMu[n/#]*3^#&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 01 2015 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,3],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n<1,n==0,sumdiv(n,d,moebius(n/d)*3^d)/n)

a(n) = (1/n)*Sum_{d|n} mu(d)*3^(n/d).
(1 - 3*x) = Product_{n>0} (1 - x^n)^a(n).
G.f.: k = 3, 1 - Sum_{i >= 1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) ~ 3^n / n. - Vaclav Kotesovec, Jul 01 2018
a(n) = 2*A046211(n) + A046209(n). - R. J. Mathar, Oct 21 2021

A065419 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(6p^2-4p+1)/(p-1)^4).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

For comparison: Product_{n>=5} (1-(6n^2-4n+1)/(n-1)^4) = 3/32. - R. J. Mathar, Feb 25 2009

			0.30749487875832709312335448607107685302...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant C_1^(4).
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]

Cf. A065418, A027377, A065431.

$MaxExtraPrecision = 1000; digits = 99; terms = 1000; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{5, -4}, {-12, -60}, 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 17 2016 *)

prodeulerrat(1-(6*p^2-4*p+1)/(p-1)^4, 1, 5) \\ Amiram Eldar, Mar 10 2021

A sign in the definition corrected by R. J. Mathar, Feb 25 2009

A269843 Decimal expansion of Hardy-Littlewood constant C_5 = Product_{p prime > 5} 1/(1-1/p)^5 (1-5/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.4098748850882364744787812123379552778963580132549454698263363988...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.

B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.

Cf. A005597, A065418, A065419, A001692.

$MaxExtraPrecision = 800; digits = 99; terms = 800; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n - 1/5^n; LR = Join[{0, 0}, LinearRecurrence[{6, -5}, {-20, -120}, 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

prodeulerrat(1/(1-1/p)^5*(1-5/p), 1, 7) \\ Amiram Eldar, Mar 11 2021

A333586 Skewes numbers for prime n-tuples p1, p2, ..., pn, with p2 - p1 = 2.

Original entry on oeis.org

1369391, 87613571, 1172531, 21432401, 204540143441, 7572964186421

Offset: 2

Hugo Pfoertner, Mar 30 2020

a(n) is the least prime p1 starting an n-tuple of consecutive primes p1, ..., pn of minimal span pn - p1, with first gap p2 - p1 = 2, such that the difference of the occurrence count of these n-tuples and the prediction by the first Hardy-Littlewood conjecture has its first sign change. When more than one such tuple exists, the n-tuple with the lexicographically earliest sequence of gaps is chosen.
These primes are called Skewes's (or Skewes) numbers for prime k-tuples in analogy to the definition for single primes. See Tóth's article for details.
a(2) is the Skewes number for twin primes, first computed by Wolf (2011).
The minimal span s(n) = pn - p1 of the n-tuples with an initial gap of 2 is s(2) = 2, s(3) = 6, s(4) = 8, s(5) = 12, s(6) = 18, s(7) = 20, s(8) = 26.

			For n=6 two types of prime 6-tuples with first gap = 2 starting at p exist:
[p, p+2, p+6, p+8, p+12, p+18] and [p, p+2, p+8, p+12, p+14, p+18]. The first one has the lexicographically earlier sequence of gaps and is therefore chosen. The Hardy-Littlewood prediction for the number of such 6-tuples with p <= P is (C_6*15^5/2^13)*Integral_{x=2..P} 1/log(x)^6 dx with C_6 given in A269846. The 15049-th 6-tuple starting with a(6)=204540143441 is the first one for which n/Integral_{x=2..a(6)} 1/log(x)^6 dx = 17.29864469487 exceeds C_6*15^5/2^13 = 17.29861231158.

Hugo Pfoertner, Illustration of growth of number of 7-tuples, (2020).
László Tóth, On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood, CMST 25(3), 2019, 143-148.
Wikipedia, Skewes's number
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. A005597, A065418, A065419, A269843, A269846, A271742, A333587.

The sequence of Skewes numbers always choosing the prime n-tuplets with minimal span, irrespective of the first gap, is A210439, and its variant A332493.

Li(x, n)=intnum(t=2, n, 1/log(t)^x);
\\ a(4)
C4=0.307494878758327093123354486071076853*(27/2); \\ A065419
\\ Start at 5 to exclude "fake" 4-tuple 3, 5, 7, 11
p1=5; p2=7; p3=11; n=0; forprime(p=13, 10^9, if(p-p1==8&&p-p2==6, n++; d=n-C4*Li(4, p3); if(d>=0, print(p1, " ", n, ">", C4*Li(4, p)); break)); p1=p2; p2=p3; p3=p);
\\ a(5)
C5=(15^4/2^11)*0.409874885088236474478781212337955277896358; \\ A269843
p1=3; p2=5; p3=7; p4=11; n=0; forprime(p=13, 10^9, if(p-p1==12&&p-p2==10, n++; d=n-C5*Li(5, p4); if(d>=0, print(p1, " ", n, ">", C5*Li(5, p)); break)); p1=p2; p2=p3; p3=p4; p4=p);

Changed title and clarified definition by Hugo Pfoertner, May 11 2020

A333587 a(n) is the least prime p1 starting an n-tuple of consecutive primes p1, ..., pn of minimal span pn - p1, with first gap p2 - p1 = 4, such that the difference of the occurrence count of these n-tuples and the prediction by the first Hardy-Littlewood conjecture has its first sign change.

Original entry on oeis.org

5206837, 337867, 827929093, 216646267, 251331775687

Offset: 2

Hugo Pfoertner, Mar 30 2020

See A333586 for more information and references.
a(2) is the Skewes number for the so-called cousin primes.
The minimal span s(n) = pn - p1 of the n-tuples with an initial gap of 4 is s(2) = 4, s(3) = 6, s(4) = 10, s(5) = 12, s(6) = 16.

Wikipedia, Cousin prime
Wikipedia, Skewes's number

Cf. A023200, A333586.

\\ Computes a(3)
Li(x,n)=intnum(t=2,n,1/log(t)^x);
C3=0.635166354604271207206696591272522417342*(9/2); \\ A065418
p1=3;p2=5;n=0;forprime(p=7,10^9,if(p-p1==6&&p-p2==2,n++;d=n-C3*Li(3,p2);if(d>=0,print(p1," ",n,">",C3*Li(3,p));break));p1=p2;p2=p)

A269846 Decimal expansion of Hardy-Littlewood constant C_6 = Product_{p prime > 6} 1/(1-1/p)^6 (1-6/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.18661429735835839665692484794418833784007394494558930487172669...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.

B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.

Cf. A005597, A065418, A065419, A269843.

$MaxExtraPrecision = 1600; digits = 99; terms = 1600; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n - 1/5^n; LR = Join[{0, 0}, LinearRecurrence[{7, -6}, {-30, -210}, 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

prodeulerrat(1/(1-1/p)^6*(1-6/p), 1, 7) \\ Amiram Eldar, Mar 11 2021

A271742 Decimal expansion of Hardy-Littlewood constant C_7 = Product_{p prime > 7} 1/(1-1/p)^7 (1-7/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 17 2016

			0.3694375103864986893231907498767507770553729138930318252910123...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 86.

B. H. Mayoh, The 2nd Goldbach conjecture revisited, BIT 8 (1968) 128-133 Table 5.

Cf. A005597, A065418, A065419, A269843, A269846.

$MaxExtraPrecision = 1100; digits = 99; terms = 1000; P[n_] := PrimeZetaP[ n] - 1/2^n - 1/3^n - 1/5^n - 1/7^n; LR = Join[{0, 0}, LinearRecurrence[ {8, -7}, {-42, -336}, 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

prodeulerrat(1/(1-1/p)^7*(1-7/p), 1, 11) \\ Amiram Eldar, Mar 11 2021

A066654 Continued fraction for the Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 6, 5, 1, 2, 2, 1, 12, 1, 2, 12, 1, 6, 5, 2, 36, 4, 1, 1, 2, 1, 2, 4, 14, 4, 6, 2, 1, 12, 2, 3, 5, 28, 1, 2, 3, 3, 1, 2, 1, 24, 1, 29, 6, 1, 11, 1, 1, 2, 9, 3, 5, 5, 4, 3, 1, 5, 3, 1, 1, 1, 1, 3, 15, 4, 10, 1, 7, 1, 1, 5, 1, 6, 5, 2, 1, 1, 8, 1, 2, 1, 1, 13, 22, 1, 2, 2, 1, 2, 1, 1, 1, 6

Offset: 0

Randall L Rathbun, Jan 16 2002

Cf. A065418.

contfrac(prodeulerrat(1-(3*p-1)/(p-1)^3, 1, 5)) \\ Amiram Eldar, Mar 10 2021

A188596 Decimal expansion of Product_{primes p} (1-1/p)^(-2)*(1-(2+A102283(p))/p).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Apr 05 2011

This is the principal scale factor in an estimate of the number of primes p not exceeding N such that p^2+p+1 is also prime [Bateman-Horn].
A102283 in the definition plays the role of the Dirichlet character modulo 3.
After splitting the product into the three modulo-3 classes of primes, this constant turns out to be the product of four factors.
One factor as mentioned by Bateman and Horn is the inverse of A073010.
The second factor is 3/4 arising from the prime 3 which is the sole prime in the class == 0 (mod 3).
The third factor is product_{p == 1 (mod 3)} (1-(3p-1)/(p-1)^3) = 0.8675121817.. which is the constant C(m=3,n=1,s=3) of the arXiv preprint, basically the C(3) variant of A065418 reduced to the modulo class.
The final factor is product_{p == 2 (mod 3)} (1+1/(p^2-1)) = 1/product_{p == 2 (mod 3)} (1-1/p^2) = 1.41406439089214763.. which is the constant zeta(m=3,n=2,s=2) of the preprint and mentioned in A175646.

			Equals 1.5217315350757058188419... = 0.92003856361849186... / A073010 .

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.1, p. 86.

Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367, constant C.
H. Davenport and A. Schinzel, A note on certain arithmetical constants, Illinois Math. J. 10 (2) (1966), 181-185.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A053182.

a073010 := evalf(Pi/3/sqrt(3)) ;
Cm3n0s2 := 1-1/(3-1)^2 ;
Cm3n1s3 := 0.867512181712394919089076584762888869720269526863 ;
Zm3n2s2 := 1.4140643908921476375655018190798293799076950693931 ;
Cm3n0s2*Cm3n1s3*Zm3n2s2/a073010 ;

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
Zs[m_, n_, s_] := (w = 2; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = (s^w - s) * P[m, n, w]/w; sumz = sumz + difz; w++]; Exp[-sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[3^(5/2)*Zs[3, 1, 3]*Z[3, 2, 2]/(4*Pi), digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

More terms from Vaclav Kotesovec, Jan 16 2021

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A065419 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(6*p^2-4*p+1)/(p-1)^4).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A269843 Decimal expansion of Hardy-Littlewood constant C_5 = Product_{p prime > 5} 1/(1-1/p)^5 (1-5/p).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A333586 Skewes numbers for prime n-tuples p1, p2, ..., pn, with p2 - p1 = 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A333587 a(n) is the least prime p1 starting an n-tuple of consecutive primes p1, ..., pn of minimal span pn - p1, with first gap p2 - p1 = 4, such that the difference of the occurrence count of these n-tuples and the prediction by the first Hardy-Littlewood conjecture has its first sign change.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A269846 Decimal expansion of Hardy-Littlewood constant C_6 = Product_{p prime > 6} 1/(1-1/p)^6 (1-6/p).

Views

Author

Keywords

Examples

References

A065419 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(6p^2-4p+1)/(p-1)^4).