A085361 -id:A085361 - cp's OEIS frontend

A085291 Decimal expansion of Alladi-Grinstead constant exp(c-1), where c is given in A085361.

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

Offset: 0

Eric W. Weisstein, Jun 25 2003

Named after the Indian-American mathematician Krishnaswami Alladi (b. 1955) and the American mathematician Charles Miller Grinstead (b. 1952). - Amiram Eldar, Jun 15 2021

			0.80939402054063913071793188059409131721595399242500030424202871504...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 120-122.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B22.

K. Alladi and C. Grinstead, On the decomposition of n! into prime powers, J. Number Theory, Vol. 9, No. 4 (1977), pp. 452-458.
Simon Plouffe, Alladi-Grinstead Constant.
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant.
Wikipedia, Multiplicative partitions of factorials.

Cf. A085288, A085289, A085290, A085361.

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

$MaxExtraPrecision = 256; RealDigits[ Exp[ Sum[ N[(-1 + Zeta[1 + n])/n, 256], {n, 350}] - 1], 10, 111][[1]] (* Robert G. Wilson v, Nov 23 2005 *)

exp(suminf(n=1, (zeta(n+1)-1)/n) - 1) \\ Michel Marcus, May 19 2020

Equals exp(c-1), where c is Sum_{n>=1} (zeta(n+1) - 1)/n (cf. A085361).

Equals lim_{n->oo} (Product_{k=1..n} (k/n)*floor(n/k))^(1/n). - Benoit Cloitre, Jul 15 2022

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 24 2003

A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).

Original entry on oeis.org

1, 1, 2, 3, 8, 10, 36, 42, 128, 216, 600, 660, 3456, 3744, 9408, 18900, 61440, 65280, 279936, 295488, 1152000, 2116800, 4878720, 5100480, 31850496, 41472000, 93450240, 163762560, 568995840, 589317120, 3265920000, 3374784000, 11324620800, 19269550080, 42188636160

Offset: 0

N. J. A. Sloane, Simon Plouffe

Product floor(n/1)*floor(n/2)*floor(n/3)*...*floor(n/n).

a(n) is the number of functions f:[n]->[n] where f(x) is a multiple of x for all x in [n]. We note that there are floor[n/x] possible choices for each image of x under f. [Dennis P. Walsh, Nov 06 2014]

			For n=4 the a(4)=8 functions are given by the image sequences <1,2,3,4>, <1,4,3,4>, <2,2,3,4>, <2,4,3,4>, <3,2,3,4>, <3,4,3,4>, <4,2,3,4>, and <4,4,3,4>. [_Dennis P. Walsh_, Nov 06 2014]

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant
Index entries for sequences related to factorial numbers

Cf. A006218, A075885, A092143, A131385, A131387, A377484, A007955, A075993.

a010786 n = product $ map (div n) [1..n]
-- Reinhard Zumkeller, Feb 26 2012

[&*[n div i: i in [1..n]]: n in [1..35]]; // Vincenzo Librandi, Oct 03 2018

Maple
```
a := n -> mul( floor(n/k), k=1..n);
```

Table[Product[Floor[n/k],{k,n}],{n,40}] (* Harvey P. Dale, May 09 2017 *)

vector(50, n, prod(k=1, n, n\k)) \\ Michel Marcus, Nov 10 2014

a(n+1) = a(n)*A208449(n)/A208450(n). - Reinhard Zumkeller, Feb 26 2012
GCD(a(n), a(n+1)) = A208448(n). - Reinhard Zumkeller, Feb 26 2012
From Vaclav Kotesovec, Oct 03 2018: (Start)
log(a(n)) ~ c * (n - log(2*Pi*n)/2), where c = 0.7885...
Conjecture: c = A085361. (End)
From Ridouane Oudra, Jan 18 2025: (Start)
a(n) = Product_{k=1..n} ((k+1)/k)^floor(n/(k+1)).
a(n) = Product_{k=1..n} k^A075993(n, k).
a(n) = A092143(n)/f(n), where f(n) = Product_{k=1..n} ((floor(n/k)-1)!).
a(n) = A092143(n)/g(n), where g(n) = Product_{k=1..n} A377484(k).
a(n)/a(n-1) = A007955(n)/A377484(n). (End)

More terms from Hieronymus Fischer, Jul 08 2007
Edited by N. J. A. Sloane, Jul 05 2008 at the suggestion of Rick L. Shepherd
a(0)=1 prepended by Alois P. Heinz, Oct 30 2023

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

Original entry on oeis.org

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

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

A075885 a(n) = 1 + n + n[n/2] + n[n/2][n/3] + n[n/2][n/3][n/4] +... where [x]=floor(x).

Original entry on oeis.org

1, 2, 5, 10, 29, 46, 169, 239, 745, 1450, 4111, 5182, 27157, 33164, 84001, 186496, 610065, 713474, 3061009, 3526553, 13783421, 27380452, 63264389, 71240523, 444872761, 620729126, 1400231613, 2615011102, 9094701085, 10008828958

Offset: 0

Paul D. Hanna, Oct 16 2002

Conjecture: limit a(n)^(1/n) = L where L = 2.200161058099... is the geometric mean of Luroth expansions, where log(L) = Sum_{n>=1} log(n)/(n*(n+1)) = 0.7885305659115... (cf. A085361).
Compare the definition of a(n) to the exponential series:
exp(n) = 1 + n + n*(n/2) + n*(n/2)*(n/3) + n*(n/2)*(n/3)*(n/4) +...

			a(5) = 1 + 5 + 5[5/2] + 5[5/2][5/3] + 5[5/2][5/3][5/4] + 5[5/2][5/3][5/4][5/5] = 1 + 5 + 5*2 + 5*2*1 + 5*2*1*1 + 5*2*1*1*1 = 46.

Paul D Hanna, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Cf. A207643, A207644, A207645, A208060, A085361.

{a(n)=1+sum(m=1,n,prod(k=1,m,floor(n/k)))}
for(n=0,60,print1(a(n),", "))

a(n)=my(k=1);1+sum(m=1,n,k*=n\m) \\ Charles R Greathouse IV, Feb 20 2012

a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} floor(n/k).

A135291 Product of the nonzero exponents in the prime factorization of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 8, 8, 14, 28, 64, 64, 100, 100, 220, 396, 540, 540, 768, 768, 1152, 1944, 4104, 4104, 5280, 7920, 16560, 21528, 31200, 31200, 40768, 40768, 48608, 78120, 161280, 230400, 277440, 277440, 571200, 907200, 1108080, 1108080, 1440504, 1440504, 2019168

Offset: 0

Leroy Quet, Dec 03 2007

nonn

a(n) = A005361(n!). For n >= 2, a(n) = the number of positive divisors of n! which themselves are each divisible by every prime <= n. For p = any prime, a(p) = a(p-1). a(0)=a(1)=1 because the product of the exponents is over the empty set.

			6! = 720 has a prime factorization of 2^4 * 3^2 * 5^1. So a(6) = 4*2*1 = 8.
Also, 720 is divisible by a(6)=8 positive divisors which themselves are each divisible by every prime <= 6 (i.e., are each divisible by 2*3*5 = 30): 30, 60, 90, 120, 180, 240, 360, 720.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from G. C. Greubel)
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A005361, A000005, A049614, A085361.

A005361 := proc(n) mul( op(2,i),i=ifactors(n)[2]) ; end: A135291 := proc(n) A005361(n!) ; end: seq(A135291(n),n=0..50) ; # R. J. Mathar, Dec 12 2007
# second Maple program:
b:= proc(n) option remember; `if`(n<1, 1,
      b(n-1)+add(i[2]*x^i[1], i=ifactors(n)[2]))
    end:
a:= n-> mul(i, i=coeffs(b(n))):
seq(a(n), n=0..44);  # Alois P. Heinz, Jun 02 2025

Table[Product[FactorInteger[n! ][[i, 2]], {i, 1, Length[FactorInteger[n! ]]}], {n, 0, 50}] (* Stefan Steinerberger, Dec 05 2007 *)
Table[Times@@Transpose[FactorInteger[n!]][[2]],{n,0,50}] (* Harvey P. Dale, Aug 16 2011 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2,n\2, s*=valp(n,p)); s \\ Charles R Greathouse IV, Oct 09 2016

from math import prod
from collections import Counter
from sympy import factorint
def A135291(n): return prod(sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()) # Chai Wah Wu, Jun 02 2025

a(n) = A000005(A049614(n)). - Ridouane Oudra, Sep 02 2019

a(n) = exp((n/log(n)) * (Sum_{k=0..M} e_k/log(n)^k) + O(n/log(n)^(M+2))) for any given integer M >= 0, where e_k = k! * Sum_{j=0..k} (1/j!) * Sum_{s>=1} (log(s+1)^j/(s+1))*log(1+1/s) are constants (e_0 = A085361) (De Koninck and Verreault, 2024, p. 54, Theorem 4.4). - Amiram Eldar, Dec 10 2024

More terms from Stefan Steinerberger and R. J. Mathar, Dec 05 2007

A242624 Decimal expansion of Product_{n>1} (1-1/n)^(1/n).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 19 2014

			0.4545121805146463170328014636843273993...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.9, pp. 121-122.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A085361, A242623, A244625, A245254.

SetDefaultRealField(RealField(100)); L:=RiemannZeta();  Exp((&+[(Evaluate(L,n)-1)/(1-n): n in [2..10^3]])); // G. C. Greubel, Nov 15 2018

evalf(exp(-sum((1-Zeta(n))/(1-n), n=2..infinity)), 120); # Vaclav Kotesovec, Dec 11 2015

Exp[-NSum[(1-Zeta[n])/(1-n), {n, 2, Infinity}, NSumTerms -> 300, WorkingPrecision -> 110]] // RealDigits[#, 10, 100]& // First

default(realprecision, 100); exp(suminf(n=2, (zeta(n)-1)/(1-n))) \\ G. C. Greubel, Nov 15 2018

numerical_approx(exp(sum((zeta(k)-1)/(1-k) for k in [2..1000])), digits=100) # G. C. Greubel, Nov 15 2018

From Amiram Eldar, Feb 06 2022: (Start)
Equals exp(-A085361).
Equals 1/A245254. (End)

Mma modified and data extended by Jean-François Alcover, May 23 2014

A244109 Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 20 2014

			2.04627745285587859107017615395043619498429055873216651873269723582433...

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 = 1..1000
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. A002210, A085361. Equals twice A340440.

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

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

NSum[Log[k*(k+1)]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *)
digits = 120; RealDigits[NSum[((1-(-1)^n)*Zeta[n+1] -1)/n, {n, 1, Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

default(realprecision, 1000); s = sumalt(n=1, ((1 + (-1)^(n+1))*zeta(n+1) - 1)/n); default(realprecision, 100); print(s) \\ Vaclav Kotesovec, Dec 11 2015

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

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

Equals Sum_{k>=1} log(k*(k+1))/(k*(k+1)).
Equals A085361 + A131688. - Vaclav Kotesovec, Dec 11 2015
Equals Sum_{n >=1} ((1 + (-1)^(n+1))*zeta(n + 1) - 1)/n. - G. C. Greubel, Nov 15 2018
Equals 2*Sum_{k>=2} log(k)/(k^2-1) = 2*A340440. - Gleb Koloskov, May 02 2021
Equals -2*Sum_{k>=1} zeta'(2*k). - Vaclav Kotesovec, Jun 17 2021

Corrected by Vaclav Kotesovec, Dec 11 2015

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

A208060 a(n) = 1 + 2n + 2^2n[n/2] + 2^3n[n/2][n/3] + 2^4n[n/2][n/3][n/4] + ... where [x]=floor(x).

Original entry on oeis.org

1, 3, 13, 43, 233, 611, 4405, 10515, 64145, 218755, 1215821, 2689083, 28162105, 61179795, 307475813, 1236997051, 8042542625, 17101581699, 146671231501, 309740445795, 2415132010441, 8877053064643, 40919003272005, 85564885298027, 1068638260341937, 2783025471994851

Offset: 0

Paul D. Hanna, Feb 23 2012

nonn

Compare the definition of a(n) to the exponential series:
exp(2*n) = 1 + 2*n + 2^2*n*(n/2) + 2^3*n*(n/2)*(n/3) + 2^4*n*(n/2)*(n/3)*(n/4) + ...
Conjecture: limit a(n)^(1/n) = 2*L where L = 2.200161058099... is the geometric mean of Luroth expansions, where log(L) = Sum_{n>=1} log(n)/(n*(n+1)) = 0.7885305659115... (cf. A085361).

			a(5) = 1 + 2*5+ 4*5[5/2] + 8*5[5/2][5/3] + 16*5[5/2][5/3][5/4] + 32*5[5/2][5/3][5/4][5/5] = 1 + 2*5 + 4*5*2 + 8*5*2*1 + 16*5*2*1*1 + 32*5*2*1*1*1 = 611.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A075885.

{a(n)=1+sum(m=1, n, prod(k=1, m, 2*floor(n/k)))}

/* More efficient: variant of a program by Charles R Greathouse IV */
{a(n)=my(k=1); 1+sum(m=1, n, k*=2*(n\m))}
for(n=0, 60, print1(a(n), ", "))

a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} 2^k*floor(n/k).

cp's OEIS Frontend

A085291 Decimal expansion of Alladi-Grinstead constant exp(c-1), where c is given in A085361.

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A010786 Floor-factorial numbers: a(n) = Product_{k=1..n} floor(n/k).

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

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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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A075885 a(n) = 1 + n + n*[n/2] + n*[n/2]*[n/3] + n*[n/2]*[n/3]*[n/4] +... where [x]=floor(x).

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A135291 Product of the nonzero exponents in the prime factorization of n!.

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

A075885 a(n) = 1 + n + n[n/2] + n[n/2][n/3] + n[n/2][n/3][n/4] +... where [x]=floor(x).

A208060 a(n) = 1 + 2n + 2^2n[n/2] + 2^3n[n/2][n/3] + 2^4n[n/2][n/3][n/4] + ... where [x]=floor(x).