A065419 -id:A065419 - cp's OEIS frontend

A065431 Continued fraction expansion of Hardy-Littlewood constant (A065419) Product_{p prime >= 5} (1-(6p^2-4p+1)/(p-1)^4).

0, 3, 3, 1, 29, 4, 1, 20, 1, 2, 1, 45, 1, 1, 1, 5, 2, 1, 6, 1, 7, 9, 1, 12, 5, 4, 1, 22, 8, 1, 1, 1, 4, 1, 2, 1, 1, 13, 2, 2, 1, 20, 1, 1, 6, 1, 4, 5, 8, 12, 1, 1, 1, 18, 1, 1, 1, 11, 1, 1813, 2, 1, 6, 2, 1, 517, 1, 1, 4, 3, 6, 1, 4, 1, 1, 7, 4, 24, 3, 5, 1, 5, 2, 4, 1, 24, 4, 2, 7, 9, 1, 59, 3, 1, 2

Offset: 0

Robert G. Wilson v, Nov 16 2001

Gerhard Niklasch, Some number-theoretical constants: 1000-digit values.

Cf. A065419 (decimal expansion).

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

Sign in definition corrected by R. J. Mathar, Feb 25 2009

Offset changed by Andrew Howroyd, Jul 04 2024

A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

Original entry on oeis.org

1, 4, 6, 20, 60, 204, 670, 2340, 8160, 29120, 104754, 381300, 1397740, 5162220, 19172790, 71582716, 268431360, 1010580540, 3817733920, 14467258260, 54975528948, 209430785460, 799644629550, 3059510616420

Offset: 0

N. J. A. Sloane

Apart from initial terms, exponents in expansion of A065419 as a product zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 4 colors. - Herbert Kociemba, Nov 25 2016

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..1666 (terms 0..200 from T. D. Noe)
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 3.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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

Cf. A001037, A027376, A054719, A054660, A054661.

Column k=4 of A074650.

A027377 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s/n); fi; end;

a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}] / n; a[0] = 1; Table[a[n], {n, 0, 23}](* Jean-François Alcover, Nov 29 2011 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,4],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n,sumdiv(n,d,moebius(d)<<(2*n/d))/n,1) \\ Charles R Greathouse IV, Nov 29 2011

a(n) = Sum_{d|n} mu(d)*4^(n/d)/n.
G.f.: k=4, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) = A054661(n) + 3 * A054660(n). - Andrey Zabolotskiy, Dec 17 2020
a(n) = 2 * (A054664(n) + A054660(n)). - Andrey Zabolotskiy, Dec 19 2020
a(n) = A054719(n)/n, n>0. - R. J. Mathar, Dec 16 2024

A065418 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

For comparison: Product_{n>=5} (1-(3n-1)/(n-1)^3) = 3/8 . - R. J. Mathar, Feb 25 2009

			0.635166354604271207206696591272522417342...

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^(3).
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. A066654, A065419, A027376.

$MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n] - 1/2^n - 1/3^n; LR = Join[{0, 0}, LinearRecurrence[{4, -3}, {-6, -24}, 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-(3*p-1)/(p-1)^3, 1, 5) \\ Amiram Eldar, Mar 10 2021

The constant equals Product_{n>=2} (zeta(n)*(1-2^-n)*(1-3^-n))^-A027376(n). - Michael Somos, Apr 05 2003

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

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

A374831 Decimal expansion of Product_{p prime} (1 - (1/(p(p - 1)))p^2/(p^2 + 1)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 21 2024

			0.4589374985054359613063426181...

Rafael Jakimczuk, Numbers of the form p-1 where p is prime, July 2024. See p. 23.

Cf. A005596, A005597, A065414, A065418, A065419, A374830 (lower bound).

PARI
```
prodeulerrat(1-p^2/(p*(p-1)*(p^2+1)))
```

A061642 Decimal expansion of Hardy-Littlewood constant for prime quadruples.

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jun 13 2001

Computed by Robert Harley.

			4.151180863237415757165285561959537515799410019333963032027163...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

Steven R. Finch, Hardy-Littlewood Constants. [Broken link]
Steven R. Finch, Hardy-Littlewood Constants. [From the Wayback machine]
Warut Roonguthai, Large Prime Quadruplets, NMBRTHRY Archives.
Eric Weisstein's World of Mathematics, Prime Quadruplet.

Cf. A065419 (constant without factor 27/2), A333586, A333587.

$MaxExtraPrecision = 1500; digits = 105; terms = 1500; 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]]; (27/2)* 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 16 2016 *)

(27/2) * prodeulerrat((p^3)*(p-4)/((p-1)^4), 1, 5) \\ Amiram Eldar, Mar 12 2021

Equals (27/2) * Product_{p prime > 3} (p^3)*(p-4)/((p-1)^4) using 27/2 = (3*(11+13)+(17+19))/4. - Frank Ellermann, Mar 31 2020

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A065431 Continued fraction expansion of Hardy-Littlewood constant (A065419) Product_{p prime >= 5} (1-(6*p^2-4*p+1)/(p-1)^4).

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A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

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A065418 Decimal expansion of Hardy-Littlewood constant Product_{p prime >= 5} (1-(3*p-1)/(p-1)^3).

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

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

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

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

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A065431 Continued fraction expansion of Hardy-Littlewood constant (A065419) Product_{p prime >= 5} (1-(6p^2-4p+1)/(p-1)^4).

A374831 Decimal expansion of Product_{p prime} (1 - (1/(p(p - 1)))p^2/(p^2 + 1)).