A001692 -id:A001692 - cp's OEIS frontend

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

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

A032164 Number of aperiodic necklaces of n beads of 6 colors; dimensions of free Lie algebras.

Original entry on oeis.org

1, 6, 15, 70, 315, 1554, 7735, 39990, 209790, 1119720, 6045837, 32981550, 181394535, 1004668770, 5597420295, 31345665106, 176319264240, 995685849690, 5642219252460, 32071565263710, 182807918979777

Offset: 0

Christian G. Bower

From Petros Hadjicostas, Aug 31 2018: (Start)
For each m >= 1, the CHK[m] transform of sequence (c(n): n>=1) has generating function B_m(x) = (1/m)*Sum_{d|m} mu(d)*C(x^d)^{m/d}, where C(x) = Sum_{n>=1} c(n)*x^n is the g.f. of (c(n): n >= 1). As a result, the CHK transform of sequence (c(n): n >= 1) has generating function B(x) = Sum_{m >= 1} B_m(x) = -Sum_{n >= 1} (mu(n)/n)*log(1 - C(x^n)).
For n, k >= 1, let a_k(n) = number of aperiodic necklaces of n beads of k colors. We then have (a_k(n): n >= 1) = CHK(c_k(n): n >= 1), where c_k(1) = k and c_k(n) = 0 for all n >= 2, with g.f. C_k(x) = Sum_{n >= 1} c_k(n)*x^n = k*x. The g.f. of (a_k(n): n >= 1) is A_k(x) = Sum_{n >= 1} a_k(n)*x^n = -Sum_{n >= 1} (mu(n)/n)*log(1-k*x^n), which is Herbert Kociemba's general formula below (except for the initial term a_k(0) = 1).
For the current sequence, k = 6.
(End)

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Seiichi Manyama, Table of n, a(n) for n = 0..1289 (terms 0..200 from T. D. Noe)
C. G. Bower, Transforms (2)
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
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 6 of A074650.
Cf. A001037, A001692 (5 colors).
Cf. A054721.

f[d_] := MoebiusMu[d]*6^(n/d)/n; a[n_] := Total[f /@ Divisors[n]]; a[0] = 1; Table[a[n], {n, 0, 20}](* Jean-François Alcover, Nov 07 2011 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,6],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n) = if (n==0, 1, sumdiv(n, d, moebius(d)*6^(n/d)/n)); \\ Michel Marcus, Dec 01 2015

"CHK" (necklace, identity, unlabeled) transform of 6, 0, 0, 0...
a(n) = Sum_{d|n} mu(d)*6^(n/d)/n, for n>0.
G.f.: k=6, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

A059861 a(n) = Product_{i=2..n} (prime(i) - 2).

Original entry on oeis.org

1, 1, 3, 15, 135, 1485, 22275, 378675, 7952175, 214708725, 6226553025, 217929355875, 8499244879125, 348469040044125, 15681106801985625, 799736446901266875, 45584977473372211875, 2689513670928960500625

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Arises in Hardy-Littlewood k-tuple conjecture. Also a(n) is the exact number of d=2 and also d=4 differences in dRRS[modulus=n-th primorial]; see A049296 (dRRS[m]=set of first differences of reduced residue system modulo m).
For n>1 this is the determinant of the (n-1) X (n-1) matrix whose diagonal is A006093(n) = {1, 2, 4, 6, 10, 12, 16, 18..} = the first primes minus 1 and all other elements are 1's. The determinant begins: / (2-1) 1 1 1 1 1 1 ... / 1 (3-1) 1 1 1 1 1 ... / 1 1 (5-1) 1 1 1 1 ... / 1 1 1 (7-1) 1 1 1 ... / 1 1 1 1 (11-1) 1 1 ... / 1 1 1 1 1 (13-1) 1 ... - Alexander Adamchuk, May 21 2006
From Gary W. Adamson, Apr 21 2009: (Start)
Equals (-1)^n * (1, 1, 1, 3, 15, ...) dot (1, -2, 4, -6, 10, ...).
a(6) = 135 = (1, 1, 1, 3, 15) dot (1, -2, 4, -6, 10) = (1, -2, 4, -18, 150). (End)

			n=4, a(4) = 1*(3-2)*(5-2)*(7-2) = 15. 48 first terms of A049296 give one complete period of dRRS[210], in which 15 d=2, 15 d=4 and 18 larger differences occur. For n=1, 2, ..., 5 in the periods of length {1, 2, 8, 48, 480, ...} [see A005867] the number of d=2 and also d=4 differences is {1, 1, 3, 15, 135, ..}

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, Sections A8, A1.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954.

A.H.M. Smeets, Table of n, a(n) for n = 1..100
Steven Brown, Distance between consecutive elements of the multiplicative group of integers modulo n, arXiv:2311.06873 [math.NT], 2023. See Table 1 p. 25.
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly, 66 (1959), 375-384.

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865, A040976.

Cf. A067549, A006093.

Table[ Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ], {n, 2, 20} ] (* Alexander Adamchuk, May 21 2006 *)
Table[Product[Prime@k - 2, {k, 2, n}], {n, 1, 18}] (* Harlan J. Brothers, Jul 02 2018 *)
a[1] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 2);
Table[a[n], {n, 18}]  (* Harlan J. Brothers, Jul 02 2018 *)
Join[{1},FoldList[Times,Prime[Range[2,20]]-2]] (* Harvey P. Dale, Apr 19 2023 *)

a(n) = prod(i=2, n, prime(i)-2); \\ Michel Marcus, Apr 16 2017

a(n) = Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ] for n>1. - Alexander Adamchuk, May 21 2006
a(n) = a(n-1) * (A000040(n) - 2) for n > 1. - A.H.M. Smeets, Dec 14 2019
a(n) = |{r | 0 <= r < primorial(n) and gcd(r, primorial(n)) = 1 and gcd(r + 2, primorial(n)) = 1}|. - Greg Tener, Oct 22 2021

Offset corrected by A.H.M. Smeets, Dec 14 2019

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

A065472 Decimal expansion of Product_{p prime} (1 - 1/(p+1)^2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

The probablity that two randomly chosen squarefree numbers are coprime. - Amiram Eldar, Aug 04 2020

The asymptotic mean of A001157(n)/(n*A000203(n)). - Richard R. Forberg, May 27 2023

			0.7758835100038954996204042844279...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
László Tóth, The unitary analogue of Pillai's arithmetical function, Collectanea Mathematica, Vol. 40, No. 1 (1989), pp. 19-30.

Cf. A000203, A001157, A008683, A038063, A078091, A116393, A145388.

digits = 98; Exp[NSum[(-1)^n*(2^(n-1)-2)*PrimeZetaP[n-1]/(n-1), {n, 3, Infinity}, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p+1)^2) \\ Amiram Eldar, Mar 17 2021

Equals lim_{n->oo} (Pi^2/(3*n^2*log(n))) * Sum_{k=1..n} A145388(k). - Amiram Eldar, May 14 2019

Equals Sum_{k>=1} mu(k)/sigma(k)^2, where mu is the Möbius function (A008683) and sigma(k) is the sum of divisors of k (A000203). - Amiram Eldar, Jan 14 2022

Definition corrected by Dan Asimov, Apr 15 2006

A065485 Decimal expansion of Murata's constant Product_{p prime} (1 + 1/(p-1)^2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001; edited Sep 16 2007 at the suggestion of R. J. Mathar

			2.8264199970675915755463917472369537490...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.4 and 2.7, pp. 106, 117.

Leo Murata, On the magnitude of the least prime primitive root, Journal of Number Theory, Vol. 37, No. 1 (1991), pp. 47-66.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Murata's Constant.
Eric Weisstein's World of Mathematics, Prime Products.

Cf. A000010, A000720, A008683, A078072, A241194, A241195.

digits = 99; terms = 1000; $MaxExtraPrecision = 500; r[n_Integer] := 2 - (1-I)^(n+1) - (1+I)^(n+1); NSum[r[n-1]*PrimeZetaP[n]/n, {n, 2, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10] // Exp // RealDigits[ #, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 + 1/(p-1)^2) \\ Vaclav Kotesovec, Sep 19 2020

Equals lim_{k->oo} (1/pi(k)) * Sum_{p prime, p <= k} (p-1)/phi(p-1), where pi(k) = A000720(k) and phi(k) = A000010(k) (Murata, 1991). - Amiram Eldar, Jul 31 2020

Equals Sum_{k>=1} mu(k)^2/phi(k)^2, where mu is the Möbius function (A008683) and phi is the Euler totient function (A000010). - Amiram Eldar, Jan 14 2022

A054720 Number of 5-ary sequences with primitive period n.

Original entry on oeis.org

1, 5, 20, 120, 600, 3120, 15480, 78120, 390000, 1953000, 9762480, 48828120, 244124400, 1220703120, 6103437480, 30517574880, 152587500000, 762939453120, 3814695297000, 19073486328120, 95367421874400, 476837158124880, 2384185742187480, 11920928955078120

Offset: 0

N. J. A. Sloane, Apr 20 2000

nonn

Equivalently, output sequences with primitive period n from a simple cycling shift register.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Column k=5 of A143324.

with(numtheory):
a:= n-> `if`(n=0, 1, add(mobius(d)*5^(n/d), d=divisors(n))):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 21 2012

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*5^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 11 2014 *)

Sum mu(d)*5^(n/d); d|n.
a(0) = 1, a(n) = n * A001692(n).
G.f.: 1 + 5 * Sum_{k>=1} mu(k) * x^k / (1 - 5*x^k). - Ilya Gutkovskiy, Apr 14 2021

A065415 Decimal expansion of Product_{p prime} (1-1/(p^4-p^3)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.85654044485354217442616798413595388...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 105.

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant A_1^(3) table 3.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A065414, A065416.

digits = 99; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, -1, 0, 1, -1}, {0, 0, 0, 4, 5, 6}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 3, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m-dm], 10, digits][[1]], Print[m]; m = m+dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

prodeulerrat(1-1/(p^4-p^3)) \\ Amiram Eldar, Mar 13 2021

A065416 Decimal expansion of Product_{p prime} (1-1/(p^5-p^4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 15 2001

			0.93126518416000433438923720555067698...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4, p. 105.

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A005596, A065414, A065415.

digits = 99; $MaxExtraPrecision = 400; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, -1, 0, 0, 1, -1}, {0, 0, 0, 0, 5, 6}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 5, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m-dm], 10, digits][[1]], Print[m]; m = m+dm]; RealDigits[s[m], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

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

A065469 Decimal expansion of Product_{p prime} (1 - 1/(p^2-1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.53071182047204479497294377247...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Cf. A000010, A000203, A007434, A008683, A013661, A065474, A072101.

$MaxExtraPrecision = 800; digits = 98; terms = 800; P[n_] := PrimeZetaP[n]; LR = LinearRecurrence[{0, 3, 0, -2}, {0, 0, -2, 0}, 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 18 2016 *)

prodeulerrat(1 - 1/(p^2-1)) \\ Amiram Eldar, Mar 13 2021

Product of A013661 by A065474. - R. J. Mathar, Mar 26 2011
From Amiram Eldar, Jan 14 2022: (Start)
Equals Sum_{k>=1} mu(k)/(phi(k)*sigma(k)), where mu is the Möbius function (A008683), phi is the Euler totient function (A000010) and sigma(k) is the sum of divisors of k (A000203).
Equals Sum_{k>=1} mu(k)/J_2(k), where J_2 is Jordan's totient function (A007434). (End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A032164 Number of aperiodic necklaces of n beads of 6 colors; dimensions of free Lie algebras.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A059861 a(n) = Product_{i=2..n} (prime(i) - 2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

A065472 Decimal expansion of Product_{p prime} (1 - 1/(p+1)^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A065485 Decimal expansion of Murata's constant Product_{p prime} (1 + 1/(p-1)^2).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A054720 Number of 5-ary sequences with primitive period n.

Views

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