A131688 -id:A131688 - cp's OEIS frontend

A027423 Number of divisors of n!.

1, 1, 2, 4, 8, 16, 30, 60, 96, 160, 270, 540, 792, 1584, 2592, 4032, 5376, 10752, 14688, 29376, 41040, 60800, 96000, 192000, 242880, 340032, 532224, 677376, 917280, 1834560, 2332800, 4665600, 5529600, 7864320, 12165120, 16422912

Offset: 0

Glen Burch (gburch(AT)erols.com), Leroy Quet

It appears that a(n+1)=2*a(n) if n is in A068499. - Benoit Cloitre, Sep 07 2002
Because a(0) = 1 and for all n > 0, 2*a(n) >= a(n+1), the sequence is a complete sequence. - Frank M Jackson, Aug 09 2013
Luca and Young prove that a(n) divides n! for n >= 6. - Michel Marcus, Nov 02 2017

			a(4) = 8 because 4!=24 has precisely eight distinct divisors: 1, 2, 3, 4, 6, 8, 12, 24.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Daniel Berend and J. E. Harmse, Gaps between consecutive divisors of factorials, Ann. Inst. Fourier, 43 (3) (1993), 569-583.
Paul Erdős, S. W. Graham, Alexsandr Ivić, and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, ed. by B. C. Berndt, H. G. Diamond, A. J. Hildebrand, Birkhauser 1996, pp. 337-355.
Florian Luca and Paul Thomas Young, On the number of divisors of n! and of the Fibonacci numbers, Glasnik Matematicki, Vol. 47, No. 2 (2012), 285-293. DOI: 10.3336/gm.47.2.05.
John D. Mahony, The next number in the sequence, Math. Gaz. (2024) Vol. 108, Issue 573, 399-406.
Wikipedia, Complete sequence.
Index entries for sequences related to factorial numbers
Index entries for sequences related to divisors of numbers

Cf. A000005, A000142, A062569, A131688, A161466 (divisors of 10!).

a027423 n = f 1 $ map (\p -> iterate (* p) p) a000040_list where
   f y ((pps@(p:_)):ppss)
     | p <= n = f (y * (sum (map (div n) $ takeWhile (<= n) pps) + 1)) ppss
     | otherwise = y
-- Reinhard Zumkeller, Feb 27 2013
(Python 3.8+)
from math import prod
from collections import Counter
from sympy import factorint
def A027423(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()) # Chai Wah Wu, Jun 25 2022

Maple
```
A027423 := n -> numtheory[tau](n!);
```

Table[ DivisorSigma[0, n! ], {n, 0, 35}]

for(k=0,50,print1(numdiv(k!),", ")) \\ Jaume Oliver Lafont, Mar 09 2009

a(n)=my(s=1,t,tt);forprime(p=2,n,t=tt=n\p; while(tt, t+=tt\=p); s*=t+1); s \\ Charles R Greathouse IV, Feb 08 2013

a(n) <= a(n+1) <= 2*a(n) - Benoit Cloitre, Sep 07 2002
From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 28 2009: (Start)
Assume, p1,p2...pm are the prime numbers less than or equal to n.
Then, a(n) = Product_{i=1..m} (bi+1), where bk = Sum_{i=1..m} floor(n/pk^i).
For example, if n=5, p1=2,p2=3,p3=5;
b1=floor(5/2)+floor(5/2^2)+floor(5/2^3)+...=2+1+0+..=3 similarly, b2=b3=1;
Thus a(5)=(3+1)(1+1)(1+1)=16. (End)
a(n) = A000005(A000142(n)). - Michel Marcus, Sep 13 2014
a(n) ~ exp(c * n/log(n) + O(n/log(n)^2)), where c = A131688 (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

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

A345682 a(n) = n! * Sum_{k=1..n} 1/(k*floor(n/k)).

Original entry on oeis.org

1, 2, 7, 26, 148, 804, 6228, 47424, 441936, 4288320, 50437440, 560373120, 7723935360, 106618256640, 1614841401600, 25127582054400, 446784010444800, 7727747269939200, 152873884406476800, 2966599550251008000, 62987912790921216000, 1378192085174919168000

Offset: 1

Vaclav Kotesovec, Jun 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..449
Vaclav Kotesovec, Plot of a(n)/n! for n = 1..1000000

Cf. A006218, A010786, A024917, A092143, A117871, A345683, A345684.

Table[n! * Sum[1/(k*Floor[n/k]), {k, 1, n}], {n, 1, 25}]
Table[n! * Sum[(HarmonicNumber[Floor[n/j]] - HarmonicNumber[Floor[n/(1 + j)]])/j, {j, 1, n}], {n, 1, 25}]

a(n) = n!*sum(k=1, n, 1/(k*(n\k))); \\ Michel Marcus, Jun 24 2021

my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (1-x^k)*log(1-x^k)/k)/(1-x))) \\ Seiichi Manyama, Jul 23 2022

a(n) ~ c * n!, where c = Sum_{j>=1} log(1 + 1/j)/j = A131688 = 1.25774...

E.g.f.: -(1/(1-x)) * Sum_{k>0} (1 - x^k) * log(1 - x^k)/k. - Seiichi Manyama, Jul 23 2022

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

A242623 Decimal expansion of Product_{n>1} (1+1/n)^(1/n).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 19 2014

			1.758743627951184824699896849661932...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.9 p. 122.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A131688, A242624, A244625.

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

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

Exp[NSum[((-1)^n*(-1 + Zeta[n]))/(n - 1), {n, 2, Infinity}, NSumTerms -> 300, WorkingPrecision -> 105] ] // RealDigits[#, 10, 103]& // First (* edited by Jean-François Alcover, May 23 2014 *)

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

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

Equals exp(A131688)/2.

Data extended by Jean-François Alcover, May 23 2014

A270859 Decimal expansion of Sum_{n >= 1} |G_n|/n^2, where G_n are Gregory's coefficients.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 24 2016

Gregory's coefficients (A002206 and A002207) are also known as (reciprocal) logarithmic numbers, Bernoulli numbers of the second kind and Cauchy numbers of the first kind. First few coefficients are G_1=+1/2, G_2=-1/12, G_3=+1/24, G_4=-19/720, etc.

			0.5290529699404390240722939394755897280940381716959625...

Bernard Candelpergher, Ramanujan summation of divergent series, Berlin: Springer, 2017. See p. 105, eq. (3.23).

Iaroslav V. Blagouchine and Marc-Antoine Coppo, A note on some constants related to the zeta-function and their relationship with the Gregory coefficients, The Ramanujan Journal, Vol. 47 (2018), pp. 457-473. See p. 470, eq. (37); arXiv preprint, arXiv:1703.08601 [math.NT], 2017.
Mümün Can, Ayhan Dil, Levent Kargin, Mehmet Cenkci and Mutlu Güloglu, Generalizations of the Euler-Mascheroni constant associated with the hyperharmonic numbers, arXiv:2109.01515 [math.NT], 2021.

Cf. A001620, A002206, A002207, A013661, A082633, A131688, A195189, A269330, A270857.

evalf(int((-Li(1-x)+gamma+ln(x))/x, x = 0..1), 150)

N[Integrate[(-LogIntegral[1 - x] + EulerGamma + Log[x])/x, {x, 0, 1}], 150]

Equals Integral_{x=0..1} (-li(1-x) + gamma + log(x))/x dx, where li(x) is the logarithmic integral.

Equals A131688 + gamma_1 + gamma^2/2 - zeta(2)/2, where gamma_1 = A082633 and gamma = A001620 (Candelpergher, 2017; Blagouchine and Coppo, 2018). - Amiram Eldar, Mar 18 2024

A321943 Decimal expansion of Ni_1 = (1/2)(gamma - log(2Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 12 2018

This constant links Euler's constant and Pi to the values of the Riemann zeta function at positive integers (see formulas).

			0.369669299246093688522926308635583575659682194332178386585...

D. Suryanarayana, Sums of Riemann zeta function, Math. Student, 42 (1974), 141-143.

Stefano Spezia, Table of n, a(n) for n = 0..10000
B. Candelpergher, Ramanujan summation of divergent series, HAL Id : hal-01150208; Lecture Notes in Math. Series (Springer), 2185, (2017), 93.
Marc-Antoine Coppo, A note on some alternating series involving zeta and multiple zeta values, Journal of Mathematical Analysis and Applications Volume 475, Issue 2, 15 July 2019, Pages 1831-1841; Preprint, , 2018.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 378.
R. J. Singh and V. P. Verma, Some series involving Riemann zeta function, Yokohama Math. J. 31 (1983), 1-4.
H. M. Srivastava, Sums of certain series of the Riemann zeta function, J. Math. Anal. App. 134 (1988), 129-140.

Cf. A001620 (Euler's constant), A000796 (Pi).

Cf. A193546, A000290, A194506, A131688.

Digits := 100; evalf((1/2)*(gamma-ln(2*Pi))+1);

First[RealDigits[N[(1/2)*(EulerGamma-Log[2*Pi])+1, 100], 10]]

PARI
```
(1/2)*(Euler-log(2*Pi))+1
```

from mpmath import *
mp.dps = 100; mp.pretty = True
+(1/2)*(euler-log(2*pi))+1

Ni_1 = Sum_{k>=2} (-1)^k*zeta(k)/(k+1).
Ni_1 = Sum_{n>0} (Integral_{x=0..1} x^2*(1-x)_{n-1} dx)/(n*n!), where (z)_n = z*(z+1)*(z+2)*...*(z+n-1) is the Pochhammer symbol.
Ni_1 = Sum_{n>=0} A193546(n)/(A000290(n + 1)*A194506(n)).

A075887 a(n) = 1 + n + n[n/2] + n[n/2][n/3] +... + n[n/2][n/3]...[n/n], where [x]=ceiling(x).

Original entry on oeis.org

1, 2, 5, 16, 45, 171, 421, 1968, 4553, 19225, 57261, 226854, 496309, 3136420, 6764563, 24850336, 84877201, 380461599, 805949533, 4411165990, 9288196621, 48275465722, 154143694937, 527401107276, 1100708161081, 8151403215501

Offset: 0

Paul D. Hanna, Oct 17 2002

a(n) ~ L^n where L = 3.517487255902369649399793699323864170685620..., with log(L) = Sum_{k=1..inf} log(k+1)/(k*(k+1)) = 1.2577468869443696300... (cf. A131688).

			a(5) = 171 = 1 +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*3 + 5*3*2 + 5*3*2*2 + 5*3*2*2*1, here [x]=ceiling(x).

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A131385, A131688.

[1] cat [1 + (&+[(&*[Ceiling(n/k): k in [1..j]]): j in [1..n]]): n in [1..50]]; // G. C. Greubel, Oct 11 2018

Table[1 +Sum[Product[Ceiling[n/k], {k,1,j}], {j,1,n}], {n,0,50}] (* G. C. Greubel, Oct 11 2018 *)

{a(n) = 1 + sum(m=1,n,prod(k=1,m,ceil(n/k)))}
for(n=0,40,print1(a(n),", "))

a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} ceiling(n/k) for n>0 and a(0)=1.

A345385 Decimal expansion of Sum_{k>=2} zeta''(k).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 17 2021

			2.3363131760589332919965154813777122447557697943302707967767827087829289...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
Michael I. Shamos, A catalog of the real numbers 2011, p. 355.

Cf. A085361, A131688, A345384.

evalf(Sum(Zeta(2, k), k = 2..infinity), 120);

RealDigits[Sum[Zeta''[k], {k, 2, 1000}], 10, 110][[1]]

PARI
```
suminf(k=2, zeta''(k))
```

Equals Sum_{k>=1} log(k+1)^2 / (k*(k+1)).

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