A143301 -id:A143301 - cp's OEIS frontend

A246849 Decimal expansion of 1-delta_0, where delta_0 is the Hall-Montgomery constant (A143301).

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

Offset: 0

Jean-François Alcover, Sep 05 2014

This constant, by coincidence, is also a limiting probability concerning the number of cycles of a given length in a random permutation.

One has P_1(xi) = 1-delta_0 = Pi^2/6 - log(xi) - log(xi)^2 - 2*Li_2(xi), where xi = 1/(1+sqrt(e)) (see A246848 and the references).

			0.82849950685846393413956002844478789037773709577...

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 29.
Michael Lugo, The number of cycles of specified normalized length in permutations, arXiv:0909.2909 [math.CO], 2009.
Eric Weisstein's MathWorld, Hall-Montgomery Constant

Cf. A143301, A246848.

Pi^2/6 + Log[1 + Sqrt[E]] - Log[1 + Sqrt[E]]^2 - 2*PolyLog[2, 1/(1 + Sqrt[E])] // RealDigits[#, 10, 101]& // First

Pi^2/6 + log(exp(1/2)+1) - log(exp(1/2)+1)^2 - 2*polylog(2, 1/(exp(1/2)+1)) \\ Charles R Greathouse IV, Sep 08 2014

from mpmath import mp, log, exp, polylog, pi
mp.dps=102
print([int(n) for n in list(str(pi**2/6 + log(exp(1/2)+1) - log(exp(1/2)+1)**2 - 2*polylog(2, 1/(exp(1/2)+1)))[2:-1])]) # Indranil Ghosh, Jul 04 2017

Pi^2/6 + log(1 + sqrt(e)) - log(1 + sqrt(e))^2 - 2*Li_2(1/(1 + sqrt(e))), where Li_2 is the dilogarithm function.

A126689 Decimal expansion of negative of Granville-Soundararajan constant.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Feb 14 2007

For any completely multiplicative function f(n) with -1 <= f(n) <= 1, the sum f(1) + f(2) + ... + f(x) is at most (c + o(1))x, where c is this constant. Further, this bound is sharp in that for any c0 > c there are infinitely many f and arbitrarily large x giving a sum less than c0*x. - Charles R Greathouse IV, May 26 2015

Named after the British mathematician Andrew James Granville (b. 1962) and the Indian-American mathematician Kannan Soundararajan (b. 1973). - Amiram Eldar, Jun 23 2021

			-0.65699901371692786827912005688957578075547419154089...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.33, p. 206.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Antal Balog, Andrew Granville and Kannan Soundararajan, Multiplicative functions in arithmetic progressions, Annales mathématiques du Québec, Vol. 37, No. 1 (2013), pp. 3-30, see p. 10; arXiv preprint, arXiv:math/0702389 [math.NT], 2007, see p. 7.
Andrew Granville and Kannan Soundararajan, The spectrum of multiplicative functions, Annals of Mathematics, Vol. 153, No. 2 (2001), pp. 407-470, alternative link.

Cf. A143301.

Digits := 40 ; K := proc(s) 0.5+add( binomial(s,k)*(-1)^k/k*(exp(0.5*k)-1),k=1..s) ; end: A126689 := proc(smax) 1.0-log(4.0)+add(K(s)*2^(2-s)/s,s=1..smax) ; end: for smax from 0 to 2*Digits do print(A126689(smax)) ; od ; # R. J. Mathar, Feb 16 2007
read("transforms3") ; Digits := 120 : x := 1+Pi^2/3+4*dilog(exp(1/2)+1) ; x := evalf(x) ; CONSTTOLIST(x) ; # R. J. Mathar, Sep 20 2009

RealDigits[ N[ 4*PolyLog[2, -Sqrt[E]] + Pi^2/3 + 1, 105]][[1]] (* Jean-François Alcover, Nov 08 2012, after R. J. Mathar *)

1-2*log(1+exp(1/2))+4*intnum(t=1,exp(1/2),log(t)/(t+1)) \\ Charles R Greathouse IV, Apr 29 2013

from mpmath import mp, polylog, sqrt, e, pi
mp.dps=106
print([int(k) for k in list(str(4*polylog(2, -sqrt(e)) + pi**2/3 + 1)[3:-1])]) # Indranil Ghosh, Jul 03 2017

Equals 1-2*log[1+sqrt e]+4*Integral_{t=1..sqrt e}([log t]/(1+t)) dt = 1-log 4+4*Sum_{s>=1} K(s)/(s*2^s) where K(s)=Sum_{k=0..s} binomial(s,k)*(-1)^k*[exp(k/2)-1]/k. - R. J. Mathar, Feb 16 2007

Equals 1 - 2 * A143301. - Amiram Eldar, Aug 25 2020

More terms from R. J. Mathar, Feb 16 2007, Sep 20 2009

A246848 Decimal expansion of 1/(1+sqrt(e)), a constant appearing in the computation of a limiting probability concerning the number of cycles of a given length in a random permutation.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 05 2014

1/(1+sqrt(e)) is the value of x that maximizes the expression Pi^2/6 - log(x) - log(x)^2 - 2*Li_2(x), where Li_2 is the dilogarithm function.

			0.37754066879814543536109943425449152124672063469108983694...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 29.
Michael Lugo, The number of cycles of specified normalized length in permutations, arXiv:0909.2909 [math.CO]

Cf. A143301, A246849.

RealDigits[1/(1 + Sqrt[E]), 10, 101] // First

1/(1+sqrt(exp(1))) \\ Michel Marcus, Sep 05 2014

A248080 Decimal expansion of P_0(xi), the maximum limiting probability that a random n-permutation has no cycle exceeding a given length.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 14 2014

			0.098711754480714692493721308237020679933333354780844...

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 29.
Michael Lugo, The number of cycles of specified normalized length in permutations, arXiv:0909.2909 [math.CO], 2009.

Cf. A143301, A246849.

xi = 1/(1 + Sqrt[E]); P0[x_] := Log[x]^2/2 + Log[x] + PolyLog[2, x] - Pi^2/12 + 1; Join[{0}, RealDigits[P0[xi], 10, 101] // First]

from mpmath import *
mp.dps=102
xi=1/(1 + sqrt(e))
C = log(xi)**2/2 + log(xi) + polylog(2, xi) - pi**2/12 + 1
print([int(n) for n in list(str(C)[2:-1])]) # Indranil Ghosh, Jul 04 2017

(1/2)*log(1 + sqrt(e))^2 - log(1 + sqrt(e)) + Li_2(1/(1 + sqrt(e))) - Pi^2/12 + 1.

A248791 Decimal expansion of P_2(xi), the maximum limiting probability that a random n-permutation has exactly two cycles exceeding a given length.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 14 2014

			0.0727887386608213733667186633181914296889295494487...

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 29.
Michael Lugo, The number of cycles of specified normalized length in permutations, arXiv:0909.2909 [math.CO] 2009.

Cf. A143301, A246849, A248080.

xi = 1/(1 + Sqrt[E]); P2[x_] := -Pi^2/12 + (1/2)*Log[x]^2 + PolyLog[2, x]; Join[{0}, RealDigits[P2[xi], 10, 100] // First]

from mpmath import *
mp.dps=101
xi=1/(1 + sqrt(e))
C = -pi**2/12 + (1/2)*log(xi)**2 + polylog(2, xi)
print([int(n) for n in list(str(C)[2:-1])]) # Indranil Ghosh, Jul 03 2017

-Pi^2/12 + (1/2)*log(1 + sqrt(e))^2 + Li_2(1/(1 + sqrt(e))).

cp's OEIS Frontend

A246849 Decimal expansion of 1-delta_0, where delta_0 is the Hall-Montgomery constant (A143301).

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A126689 Decimal expansion of negative of Granville-Soundararajan constant.

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A246848 Decimal expansion of 1/(1+sqrt(e)), a constant appearing in the computation of a limiting probability concerning the number of cycles of a given length in a random permutation.

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A248080 Decimal expansion of P_0(xi), the maximum limiting probability that a random n-permutation has no cycle exceeding a given length.

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A248791 Decimal expansion of P_2(xi), the maximum limiting probability that a random n-permutation has exactly two cycles exceeding a given length.

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Mathematica

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