A001692 -id:A001692 - cp's OEIS frontend

A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

This is 1/Artin's constant, see A005596.

			2.67411272557002150896041183...

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

Cf. A005596, A048296, A351219.

$MaxExtraPrecision = 1200; digits = 99; terms = 1200; P[n_] := PrimeZetaP[n ]; LR = Join[{0, 0}, LinearRecurrence[{2, 0, -1}, {2, 3, 6}, term+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-p-1)) \\ Amiram Eldar, Mar 15 2021

A065178 Number of site swap patterns with 2 balls and exact period n.

Original entry on oeis.org

1, 2, 6, 15, 42, 107, 294, 780, 2128, 5781, 15918, 43885, 122010, 340323, 954394, 2685930, 7588770, 21507696, 61144062, 174283887, 498012094, 1426213191, 4092816966, 11767176070, 33890202192, 97761428205, 282424564744

Offset: 1

Antti Karttunen, Oct 19 2001

nonn

When interspersed with 0's, exponents in expansion of A065481 as a product zeta(n)^(-a(n)).

			We have one period 1 (2), two period 2 (31/13 and 40/04) and six period three 2-ball siteswaps (312, 330, 411, 420, 501, 600) (The average of the digits is always 2).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Juggling Information Service, Site Swap FAQs
M. Macauley, Braids and Juggling patterns, Thesis (Harvey Mudd College) 2003, eq. (A1).
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Row 2 of A065177.

Cf. A008683, A065179, A065180, A065481.

[seq(DistSS(p,2),p=1..60)];
A065178 := proc(n)
    add( mobius(n/d)*(3^d-2^d),d=numtheory[divisors](n)) /n ;
end proc:
seq(A065178(n),n=1..30) ; # R. J. Mathar, Aug 05 2015

a[n_] := DivisorSum[n, MoebiusMu[n/#] * (3^#-2^#)&] / n; Array[a, 30] (* Jean-François Alcover, Mar 05 2016, after R. J. Mathar *)

a(n) ~ 3^n/n. - Vaclav Kotesovec, Mar 05 2016
Inverse Euler transform of A133494. - Alois P. Heinz, Jun 23 2018
G.f.: Sum_{k>=1} mu(k) * log(1 + x^k/(1 - 3*x^k))/k. - Seiichi Manyama, Apr 14 2025

A059865 Product_{i=4..n} (prime(i) - 6).

Original entry on oeis.org

1, 1, 1, 1, 5, 35, 385, 5005, 85085, 1956955, 48923875, 1516640125, 53082404375, 1964048961875, 80526007436875, 3784722349533125, 200590284525255625, 11032465648889059375, 672980404582232621875, 43743726297845120421875

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Arises in Hardy-Littlewood prime k-tuplet conjectural formulas. Also the sequence gives the exact numbers of X42424Y difference-pattern in dRRS[m], where m=modulus=A002110(n). See A049296 (=dRRS[210]=list of first differences of reduced residue system modulo 210=4th primorial). A pattern X42424Y corresponds to a residue-sextuple or it is their difference-quintuple, X,Y > 4. Analogous pattern for primes is in A022008.

a(352) has 1001 decimal digits. - Michael De Vlieger, Mar 06 2017

			a(7) = (prime(4)-6) * (prime(5)-6) * (prime(6)-6) * (prime(7)-6) = 1 * 5* 7 *11 = 385
 Also in one period of dRRS with 2,6,30,210,2310,... modulus [A002110(n)] 1,2,8,48,480,... differences occur [A005867(n)]. The number of X42424Y residue-difference-patterns are 0,1,1,1,5,... respectively starting at suitable residues coprime to A002110(n).

See A059862 for references.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

Michael De Vlieger, Table of n, a(n) for n = 1..351
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. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

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

Table[Product[Prime@ i - 6, {i, 4, n}], {n, 19}] (* Michael De Vlieger, Mar 06 2017 *)

a(n) = prod(k=4, n, prime(k) - 6); \\ Michel Marcus, Mar 06 2017

A065476 Decimal expansion of Product_{p prime >= 3} (1 - (p+2)/p^3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.7236484022982000094088491498...

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Peter C. Sarnak, Class Numbers of Indefinite Binary Quadratic Forms II, Journal of Number Theory, Vol. 21, No. 3 (1985), pp. 333-346.
Eric Weisstein's World of Mathematics, Sarnak's Constant.

Cf. A065477, A078090.

$MaxExtraPrecision = 500; digits = 98; terms = 500; P[n_] := PrimeZetaP[n] - 1/2^n; LR = Join[{0, 0}, LinearRecurrence[{0, 1, 2}, {-2, -6, -2}, 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 - (p+2)/p^3, 1, 3) \\ Amiram Eldar, Mar 15 2021

A065489 Decimal expansion of Product_{p prime} (1 + 1/(p^2+p-1)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Nov 19 2001

			1.419562880505485919317235861789735359... = 1/0.704442200999....

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

Cf. A065463, A078077.

$MaxExtraPrecision = 1200; digits = 99; terms = 1200; P[n_] := PrimeZetaP[n ]; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {2, -3, 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 18 2016 *)

prodeulerrat(1 + 1/(p^2+p-1)) \\ Amiram Eldar, Mar 15 2021

1 divided by A065463. - R. J. Mathar, Mar 26 2011

A065466 Decimal expansion of Product_{p prime} (1 - 1/(p^3*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

			0.947733262143675375939521537654189613...

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

Cf. A078085.

$MaxExtraPrecision = 600; digits = 98; terms = 600; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0, 0}, LinearRecurrence[{-2, -1, 0, 1, 1}, {-4, 5, -6, 7, -12}, 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^3*(p+1))) \\ Amiram Eldar, Mar 13 2021

A065480 Decimal expansion of Product_{p prime} (1 - 1/(p^2+p-1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

The probability that two numbers are coprime given that they are both coprime to a randomly chosen third number. - Luke Palmer, Apr 27 2019

			0.6695802905390623676350256956124342...

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

Cf. A065592, A078081.

Cf. A013661, A065463, A065473.

digits = 98; Exp[NSum[(1/2)*(-2 + (-2)^n - ((1/2)*(-1 - Sqrt[5]))^n*(-1 + Sqrt[5]) + ((1/2)*(-1 + Sqrt[5]))^n*(1 + Sqrt[5]))*PrimeZetaP[n - 1]/(n - 1), {n, 3, Infinity}, WorkingPrecision -> 4 digits, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

prodeulerrat(1 - 1/(p^2+p-1)) \\ Amiram Eldar, Mar 14 2021

Equals A065473*zeta(2)/A065463. - Luke Palmer, Apr 27 2019

A065493 Decimal expansion of the Feller-Tornier constant (1 + A065474)/2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

The asymptotic density of numbers with an even number of non-unitary prime divisors (A333634). - Amiram Eldar, May 23 2020

Named after the Croatian-American mathematician William Feller (1906-1970) and the German mathematician Erhard Tornier (1894-1982). - Amiram Eldar, Jun 16 2021

			0.661317049469622335289765846274...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.4.1, p. 106.

Jayadev S. Athreya, Cristian Cobeli, and Alexandru Zaharescu, Visibility phenomena in hypercubes, arXiv:2204.03147 [math.NT], 2022.
Willy Feller and Erhard Tornier, Mengentheoretische Untersuchung von Eigenschaften der Zahlenreihe, Mathematische Annalen, Vol. 107 (1933), pp. 188-232.
Mizan R. Khan and Riaz R. Khan, To count clean triangles we count on imph(n), arXiv:2012.11081 [math.CO], 2020. Mentions this constant.
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Feller-Tornier Constant.
Eric Weisstein's World of Mathematics, Prime Products.
Wikipedia, Feller-Tornier constant.

Cf. A065474, A078080, A333634.

digits = 98; r[n_] := -2^n; 1/2 + (1/2) Exp[NSum[r[n]*(PrimeZetaP[2*n]/n), {n, 1, Infinity}, NSumTerms -> 1000, WorkingPrecision -> 2 digits ]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

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

A118266 Coefficient of q^n in (1-q)^5/(1-5q); dimensions of the enveloping algebra of the derived free Lie algebra on 5 letters.

Original entry on oeis.org

1, 0, 10, 40, 205, 1024, 5120, 25600, 128000, 640000, 3200000, 16000000, 80000000, 400000000, 2000000000, 10000000000, 50000000000, 250000000000, 1250000000000, 6250000000000, 31250000000000, 156250000000000

Offset: 0

Mike Zabrocki, Apr 20 2006

nonn

For n>=5, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for fixed, different x_1, x_2, x_3, x_4, x_5 in {1,2,...,n} and fixed y_1, y_2, y_3, y_ 4, y_5 in {1,2,3,4,5} we have f(x_i)<>y_i, (i=1,2,3,4,5). - Milan Janjic, May 13 2007

C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.

Michael De Vlieger, Table of n, a(n) for n = 0..1431
Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082, 2005. See also Canad. J. Math. 60 (2008), no. 2, 266-296.
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. 21.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A001692, A118264, A118265.

f:=n->add((-1)^k*binomial(5,k)*5^(n-k),k=0..min(n,4)): seq(f(i),i=0..15);

a[n_] := If[n<6, {1, 0, 10, 40, 205, 1024}[[n+1]], 1024*5^(n-5)];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2018 *)

G.f.: (1-q)^5/(1-5q) sum( (-1)^k*C(5,k) 5^(n-k); k=0..min(n,5));

a(n) = 1024*5^(n-5) for n>5. - Jean-François Alcover, Dec 10 2018

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

cp's OEIS Frontend

A065488 Decimal expansion of Product_{p prime} (1 + 1/(p^2-p-1)).

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A065178 Number of site swap patterns with 2 balls and exact period n.

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A059865 Product_{i=4..n} (prime(i) - 6).

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A065476 Decimal expansion of Product_{p prime >= 3} (1 - (p+2)/p^3).

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A065489 Decimal expansion of Product_{p prime} (1 + 1/(p^2+p-1)).

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A065466 Decimal expansion of Product_{p prime} (1 - 1/(p^3*(p+1))).

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A065480 Decimal expansion of Product_{p prime} (1 - 1/(p^2+p-1)).

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A065493 Decimal expansion of the Feller-Tornier constant (1 + A065474)/2.