A085291 -id:A085291 - cp's OEIS frontend

A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

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

Offset: 1

R. J. Mathar, Sep 14 2007

Given A131385(n) = Product_{k=1..n} floor((n+k)/k), then limit A131385(n+1)/A131385(n) = exp(c), where c = this constant. - Paul D. Hanna, Nov 26 2012

Closely related to A085361 (the exponent in the definition of A085291). - Yuriy Sibirmovsky, Sep 04 2016

			1.257746886944369630009899830495881528511540890508884868977540833522...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 62. [Jean-François Alcover, Mar 21 2013]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.
Mark W. Coffey, Series of zeta values, the Stieltjes constants and a sum S_gamma(n), arXiv:math-ph/0706.0345, 2007-2009, eq (38a).
Paul Erdős, S. W. Graham, Aleksandar Ivic and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Volume 138, Progress in Mathematics pp 337-355.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Cf. A000005, A002210, A027423, A075887, A131385, A244109, A001620, A085361, A345682.

SetDefaultRealField(RealField(100)); L:=RiemannZeta(); (&+[(-1)^(n+1)*Evaluate(L,n+1)/n: n in [1..10^3]]); // G. C. Greubel, Nov 15 2018

evalf(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Sum[ -Zeta'[1 + k], {k, 1, Infinity}] (* Vladimir Reshetnikov, Dec 28 2008 *)
Integrate[EulerGamma/x + PolyGamma[0, 1+x]/x, {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* or *) Integrate[x*Log[x]/((1-x)*Log[1-x]), {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* Jean-François Alcover, Feb 04 2013 *)
$MaxExtraPrecision = 200; NIntegrate[HarmonicNumber[t]/t, {t, 0, 1}, WorkingPrecision -> 105] (* Yuriy Sibirmovsky, Sep 04 2016 *)
digits = 120; RealDigits[NSum[(-1)^(n + 1)*Zeta[n + 1]/n, {n,1,Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

PARI
```
sumalt(s=1, (-1)^(s+1)/s*zeta(s+1) )
```

suminf(k=2, -zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

numerical_approx(sum((-1)^(k+1)*zeta(k+1)/k for k in [1..1000]), digits=100) # G. C. Greubel, Nov 15 2018

Equals Sum_{s>=1} (-1)^(s+1)*zeta(s+1)/s.
Equals Sum_{k>=1} -zeta'(1 + k), where Zeta' is the derivative of the Riemann zeta function. - Vladimir Reshetnikov, Dec 28 2008
Equals Sum_{s>=1} log(1+1/s)/s. - Jean-François Alcover, Mar 26 2013
Equals Integral_{t=0..1} H(t)/t dt. Compare to A001620 = Integral_{t=0..1} H(t) dt. Where H(t) are generalized harmonic numbers. - Yuriy Sibirmovsky, Sep 04 2016
Equals lim_{n->oo} log(d(n!))*log(n)/n, where d(n) is the number of divisors of n (A000005) (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

Extended to 105 digits by Jean-François Alcover, Feb 04 2013

A085361 Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jun 25 2003

The Alladi-Grinstead constant (A085291) is exp(c-1).

			0.78853056591150896106027632345455466647274966822328164975515640230178...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 528 and 538.
Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant.
Eric Weisstein's World of Mathematics, Convergence Improvement.

Cf. A002210, A085291, A242624, A244109, A245254.

SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[(Evaluate(L,n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018

evalf(sum((Zeta(n+1)-1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-(-1)^k*Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Sum[(-1+Zeta[1+n])/n,{n,Infinity}]
NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms ->5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)

suminf(n=1,(zeta(n+1)-1-2^(-n-1))/n)+log(2)/2 \\ Charles R Greathouse IV, Feb 20 2012

sumalt(k=2, -(-1)^k * zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021

import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps=108 #precision
mpmath.nsum(lambda n: (-1+mpmath.zeta(1+n))/n, [1,mpmath.inf]) # Peter Luschny, Jul 14 2012

numerical_approx(sum((zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018

Equals Sum_{n>=2} log(n/(n-1))/n = Sum_{n>=1, k>=2} 1/(n*k^(n+1)). [From Mathworld links]
Equals -Sum_{k>=2} (-1)^k * zeta'(k). - Vaclav Kotesovec, Jun 17 2021
Equals log(A245254) = Sum_{k>=1} log(k)/(k*(k+1)). - Amiram Eldar, Jun 27 2021
Equals -log(A242624). - Amiram Eldar, Feb 06 2022

A131385 Product ceiling(n/1)ceiling(n/2)ceiling(n/3)...ceiling(n/n) (the 'ceiling factorial').

Original entry on oeis.org

1, 1, 2, 6, 16, 60, 144, 672, 1536, 6480, 19200, 76032, 165888, 1048320, 2257920, 8294400, 28311552, 126904320, 268738560, 1470873600, 3096576000, 16094453760, 51385466880, 175814737920, 366917713920, 2717245440000, 6782244618240, 22754631352320, 69918208819200

Offset: 0

Hieronymus Fischer, Jul 08 2007

nonn

From R. J. Mathar, Dec 05 2012: (Start)
a(n) = b(n-1) because a(n) = Product_{k=1..n} ceiling(n/k) = Product_{k=1..n-1} ceiling(n/k) = n*Product_{k=2..n-1} ceiling(n/k) = Product_{k=1..1} (1+(n-1)/k)*Product_{k=2..n-1} ceiling(n/k).
The cases of the product are (i) k divides n but does not divide n-1, ceiling(n/k) = n/k = 1 + floor((n-1)/k), (ii) k does not divide n but divides n-1, ceiling(n/k) = 1 + (n-1)/k = 1 + floor((n-1)/k) and (iii) k divides neither n nor n-1, ceiling(n/k) = 1 + floor((n-1)/k).
In all cases, including k=1, a(n) = Product_{k=1..n-1} (1+floor((n-1)/k)) = Product_{k=1..n-1} floor(1+(n-1)/k) = b(n-1).
(End)
a(n) is the number of functions f:D->{1,2,..,n-1} where D is any subset of {1,2,..,n-1} and where f(x) == 0 (mod x) for every x in D. - Dennis P. Walsh, Nov 13 2015

			From _Paul D. Hanna_, Nov 26 2012: (Start)
Illustrate initial terms using formula involving the floor function []:
  a(1) = 1;
  a(2) = [2/1] = 2;
  a(3) = [3/1]*[4/2] = 6;
  a(4) = [4/1]*[5/2]*[6/3] = 16;
  a(5) = [5/1]*[5/2]*[7/3]*[8/4] = 60;
  a(6) = [6/1]*[7/2]*[8/3]*[9/4]*[10/5] = 144.
Illustrate another alternative generating method:
  a(1) = 1;
  a(2) = (2/1)^[1/1] = 2;
  a(3) = (2/1)^[2/1] * (3/2)^[2/2] = 6;
  a(4) = (2/1)^[3/1] * (3/2)^[3/2] * (4/3)^[3/3] = 16;
  a(5) = (2/1)^[4/1] * (3/2)^[4/2] * (4/3)^[4/3] * (5/4)^[4/4] = 60.
(End)
For n=3 the a(3)=6 functions f from subsets of {1,2} into {1,2} with f(x) == 0 (mod x) are the following: f=empty set (since null function vacuously holds), f={(1,1)}, f={(1,2)}, f={(2,2)}, f={(1,1),(2,2)}, and f={(1,2),(2,2)}. - _Dennis P. Walsh_, Nov 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
D. P. Walsh, Notes on finite functions from subsets of {1,2,...,n} into {1,2,...,n}

Cf. A010786, A131387, A075885, A131688, A308820, A092143.

a:= n-> mul(ceil(n/k), k=1..n):
seq(a(n), n=0..40); # Dennis P. Walsh, Nov 13 2015

Table[Product[Ceiling[n/k],{k,n}],{n,25}] (* Harvey P. Dale, Sep 18 2011 *)

a(n)=prod(k=1,n-1,floor((n+k-1)/k)) \\ Paul D. Hanna, Feb 01 2013

a(n)=prod(k=1,n-1,((k+1)/k)^floor((n-1)/k))
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Feb 01 2013

a(n) = Product_{k=1..n} ceiling(n/k).
Formulas from Paul D. Hanna, Nov 26 2012: (Start)
a(n) = Product_{k=1..n-1} floor((n+k-1)/k) for n>1.
a(n) = Product_{k=1..n-1} ((k+1)/k)^floor((n-1)/k) for n>1.
Limits: Let L = limit a(n+1)/a(n) = 3.51748725590236964939979369932386417..., then
(1) L = exp( Sum_{n>=1} log((n+1)/n) / n ) ;
(2) L = 2 * exp( Sum_{n>=1} (-1)^(n+1) * Sum_{k>=2} 1/(n*k^(n+1)) ) ;
(4) L = exp( Sum_{n>=1} (-1)^(n+1) * zeta(n+1)/n ) ;
(5) L = exp( Sum_{n>=1} log(n+1) / (n*(n+1)) ) = exp(c) where c = constant A131688.
Compare L to Alladi-Grinstead constant defined by A085291 and A085361.
(End)
a(n) = A308820(n)/A092143(n-1) for n > 0. - Ridouane Oudra, Sep 28 2024

a(0)=1 prepended by Alois P. Heinz, Oct 30 2023

A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 15 2014

The geometric mean of the Yule-Simon distribution with parameter value 1 (A383855) approaches this constant. In general, the geometric mean of the Yule-Simon distribution approaches Product_{k>=2} k^(1/(p*Beta(k,p+1))). - Jwalin Bhatt, May 12 2025

			2.200161058099026553194557866559944872685662324752723888723145117763169...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.

Cf. A002210, A085291, A085361, A242624, A244109(V), A131688(W), A383855.

evalf(exp(Sum((Zeta(n+1)-1)/n, n=1..infinity)), 120); # Vaclav Kotesovec, Dec 11 2015

Exp[NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}]] (* Vaclav Kotesovec, Dec 11 2015 *)

Equals exp(A085361).
U*V*W = 1, where V is A244109 and W is A131688.
Equals e * A085291. - Amiram Eldar, Jun 27 2021
Equals 1/A242624. - Amiram Eldar, Feb 06 2022

Corrected by Vaclav Kotesovec, Dec 11 2015

A085288 Number of sorted multiplicative partitions of n!.

Original entry on oeis.org

1, 1, 3, 3, 10, 10, 30, 75, 220, 220, 588, 588, 1568, 3696, 11616, 11616, 30492, 30492, 84700, 173250, 441000, 441000, 1262520, 2777544, 6957720, 16731660, 43506760, 43506760, 98658000, 98658000, 277101000, 541886400, 1322481600, 2715495552

Offset: 2

Eric W. Weisstein, Jun 23 2003

nonn

Number of decompositions of n! into factors of the form (p_k)^(e^k).

			4! = 3*8 = 2*3*4 = 2*2*2*3, so a(4)=3.
5! = 3*5*8 = 2*3*4*5 = 2*2*2*3*5, so a(5)=3.

Amiram Eldar, Table of n, a(n) for n = 2..6940
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Cf. A000142, A000688, A085289, A085290, A085291.

f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger[ n!]; Array[f, 34, 2] (* Robert G. Wilson v, Sep 22 2006 *)

a(n) = A000688(n!) = A000688(A000142(n)). - Vladeta Jovovic, Jun 27 2003

More terms from Vladeta Jovovic, Jun 27 2003

A085289 Number of sorted multiplicative partitions of n! of length n.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 12, 31, 31, 78, 78, 191, 418, 1220, 1220, 3015, 3015, 7889, 15413, 37012, 37012, 100559, 211426, 501520, 1157544, 2880550, 2880550, 6307387, 6307387, 17134912, 32298263, 75200323, 148928952, 359182792, 359182792

Offset: 2

Eric W. Weisstein, Jun 23 2003

nonn

If p is prime then a(p) = a(p-1). - David Wasserman, Jan 31 2005

			4! = 2*2*2*3, so a(4)=1.
6! = 2*2*2*2*5*9 = 2*2*3*3*4*5, so a(6)=2.

Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Cf. A085288, A085290, A085291.

More terms from David Wasserman, Jan 31 2005

A085290 Max[p1^b1] over all sorted multiplicative partitions of n! of length n.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 16, 16, 16, 16, 16, 16, 16

Offset: 4

Eric W. Weisstein, Jun 23 2003

nonn

			6! = 2*2*2*2*5*9 = 2*2*3*3*4*5, the smallest terms of which are 2 and 2, so a(6)=Max[2,2]=2.

Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Cf. A085288, A085289, A085291.

Cf. A103332.

works(n, m) = local(f, s, l, p, x); f = factor(n!); s = 0; l = matsize(f)[1]; for (i = 1, l, p = f[i, 1]; x = 1; while (p^x < m, x++); s += f[i, 2]\x; if (f[i, 2] < x, return(0))); s >= n; a(n) = local(f, m); f = factor(n); m = 2; while (works(n, m), m++); m - 1 \\ David Wasserman, Jan 31 2005

More terms from David Wasserman, Jan 31 2005

cp's OEIS Frontend

A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

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A085361 Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.

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A131385 Product ceiling(n/1)*ceiling(n/2)*ceiling(n/3)*...*ceiling(n/n) (the 'ceiling factorial').

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A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

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Maple

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A085288 Number of sorted multiplicative partitions of n!.

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Mathematica

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A085289 Number of sorted multiplicative partitions of n! of length n.

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A131385 Product ceiling(n/1)ceiling(n/2)ceiling(n/3)...ceiling(n/n) (the 'ceiling factorial').