A092143 -id:A092143 - cp's OEIS frontend

A117871 Decimal expansion of Sum_{i>=1} 1/A092143(i).

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

Offset: 1

Jonathan Vos Post, May 31 2007

It follows from the Mingarelli reference that this number is irrational.

			1.6917992098217123513392618067876318698236937629258191345569...

Angelo B. Mingarelli, Abstract factorials of arbitrary sets of integers, arXiv:0705.4299 [math.NT], 200-2012.

Cf. A092143.

Digits := 60 : A092143 := proc(n) option remember ; local dvs ; if n = 1 then 1 ; else dvs := numtheory[divisors](n) ; A092143(n-1)*mul(i,i=dvs) ; fi ; end: A129635 := proc(isum) a := 0.0 ; for i from 1 to isum do a := a+1.0/A092143(i) ; print(evalf(a)) ; od ; RETURN(a) ; end: A129635(200) ; # R. J. Mathar, Sep 02 2007

digits = 105; A092143[m_] := For[n = k = 1, k <= m, k++, Do[n = n*d, {d, Divisors[k]}]; If[k == m, Return[n]]] ;rd[j_] := rd[j] = RealDigits[ N[ Sum[ 1/A092143[m], {m, 1, 2^j}], digits]][[1]]; rd[j = 4]; While[ rd[j] != rd[j - 1], j++]; rd[j] (* Jean-François Alcover, Oct 30 2012 *)

More terms from R. J. Mathar, Sep 02 2007
Edited by N. J. A. Sloane, Sep 16 2007 and May 06 2008
More digits from R. J. Mathar, Jul 12 2009

A092144 a(n) = A092143(n)/n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 96, 288, 2880, 2880, 414720, 414720, 5806080, 87091200, 5573836800, 5573836800, 1805923123200, 1805923123200, 722369249280000, 15169754234880000, 333734593167360000, 333734593167360000, 4613547015945584640000, 23067735079727923200000

Offset: 1

Jon Perry, Mar 31 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..200

Cf. A092143.

[(&*[j^Floor(n/j): j in [1..n]])/Factorial(n): n in [1..40]]; // G. C. Greubel, Feb 05 2024

Table[Product[k^Floor[n/k], {k,n}]/n!, {n,40}] (* G. C. Greubel, Feb 05 2024 *)

my(z=1); for(i=1,20, fordiv(i,j,z*=j); print1(z/i!, ", "))

[product(j^(n//j) for j in range(1,n+1))//factorial(n) for n in range(1,41)] # G. C. Greubel, Feb 05 2024

Terms a(21) onward added by G. C. Greubel, Feb 05 2024

A129439 An analog of Pascal's triangle: T(n,k) = A092143(n)/(A092143(n-k)*A092143(k)), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 8, 12, 8, 1, 1, 5, 20, 20, 5, 1, 1, 36, 90, 240, 90, 36, 1, 1, 7, 126, 210, 210, 126, 7, 1, 1, 64, 224, 2688, 1680, 2688, 224, 64, 1, 1, 27, 864, 2016, 9072, 9072, 2016, 864, 27, 1, 1, 100, 1350, 28800, 25200, 181440, 25200, 28800, 1350, 100, 1

Offset: 0

Peter Bala, Apr 15 2007

It appears that the T(n,k) are always integers. This would follow from the conjectured prime factorization given in the comments section of A092143.

			Triangle starts
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1;
  1, 8, 12,  8, 1;
  1, 5, 20, 20, 5, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007955 (second column), A092143.

A092143:= func< n |n eq 0 select 1 else (&*[Factorial(Floor(n/j)): j in [1..n]]) >;
A129439:= func< n,k | A092143(n)/(A092143(k)*A092143(n-k)) >;
[A129439(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Feb 06 2024

A092143[n_]:= Product[Floor[n/j]!, {j,n}];
A129439[n_, k_]:= A092143[n]/(A092143[n-k]*A092143[k]);
Table[A129439[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 06 2024 *)

def A092143(n): return product(factorial(n//j) for j in range(1,n+1))
def A129439(n,k): return A092143(n)//(A092143(n-k)*A092143(k))
flatten([[A129439(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Feb 06 2024

T(n,k) = Product_{j=1..n} floor(n/j)!/((Product_{j=1..n-k} floor((n-k)/j)!)*(Product_{j=1..k} floor(k/j)!)).
T(n, 1) = A007955(n).
T(n, n-k) = T(n, k). - G. C. Greubel, Feb 06 2024

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

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

A345683 a(n) = n! * Sum_{k=1..n} 1/floor(n/k).

Original entry on oeis.org

1, 3, 14, 66, 444, 2880, 25080, 216720, 2247840, 24071040, 304335360, 3752179200, 54965433600, 810550540800, 13176376012800, 219079045785600, 4078723532083200, 75227891042304000, 1550619342784512000, 31871016307113984000, 710529031487987712000, 16180987966182014976000

Offset: 1

Vaclav Kotesovec, Jun 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..448
Vaclav Kotesovec, Plot of a(n) / (n*n!) for n = 1..1000000
Mathoverflow, An asymptotic formula for a sum involving powers of floor functions, 2019.

Cf. A006218, A010786, A092143, A117871, A345682, A345684, A345686, A355987.

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

a(n) = n!*sum(k=1, n, 1/(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))/(1-x))) \\ Seiichi Manyama, Jul 23 2022

from math import factorial, isqrt
def A345683(n): return (m:=factorial(n))*(n-1)+m//n+sum((q:=n//k)*(m//k-m//(k-1))+m//q for k in range(2,isqrt(n)+1)) # Chai Wah Wu, Oct 27 2023

a(n) ~ c * n * n!, where c = Sum_{j>=1} 1/(j^2*(j+1)) = Pi^2/6 - 1 = 0.644934... [proved by Harry Richman, see Mathoverflow link]

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

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

A345684 a(n) = n! * Sum_{k=1..n} k/floor(n/k).

Original entry on oeis.org

1, 5, 32, 198, 1584, 12480, 122520, 1214640, 14011200, 166924800, 2274894720, 31135104000, 485667705600, 7710089587200, 133974352512000, 2386854434764800, 46621903994265600, 918384939343872000, 19760215067873280000, 430137075045629952000, 10042411264251125760000

Offset: 1

Vaclav Kotesovec, Jun 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..448

Cf. A006218, A010786, A076000, A092143, A117871, A345682, A345683.

Table[n!*Sum[k/Floor[n/k], {k, 1, n}], {n, 1, 25}]
Table[n!*Sum[(Floor[n/j] - Floor[n/(1 + j)])*((1 + Floor[n/j] + Floor[n/(1 + j)])/2/j), {j, 1, n}], {n, 1, 25}]

a(n) = n!*sum(k=1, n, k/(n\k)); \\ Michel Marcus, Jun 23 2021

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

a(n) ~ c * n^2 * n!, where c = Sum_{j>=1} (2*j + 1) / (2*j^3*(j+1)^2) = Pi^2/12 + zeta(3)/2 - 1 = 0.423495...

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

A129364 a(n) = Product_{k = 1..n} A066841(k).

Original entry on oeis.org

1, 2, 6, 96, 480, 207360, 1451520, 2972712960, 722369249280, 5778953994240000, 63568493936640000, 9111096278347394580480000, 118444251618516129546240000, 10400352846118664303196241920000

Offset: 1

Peter Bala, Apr 11 2007

Conjecture: a(n) divides A092287(n) for all n - see comments in A129365.

Cf. A007955, A066841, A092143, A092287, A129365.

Table[Product[Floor[n/k]!^k, {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Jun 24 2021 *)
Table[Product[k^(Floor[n/k]*(1 + Floor[n/k])/2), {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Jun 24 2021 *)

a(n) = prod(k=1, n, k^((n\k) * (1 + n\k) \ 2)); \\ Daniel Suteu, Sep 12 2018

a(n) = Product_{k = 1..n} Product_{d|k} d^(k/d).
a(n) = Product_{k = 1..n} ((floor(n/k))!)^k.
a(n) = exp(Sum_{k = 1..n} log(k)/2 * floor(n/k) * floor(1 + n/k)). - Daniel Suteu, Sep 12 2018
log(a(n)) ~ c * n^2, where c = -zeta'(2)/2 = A073002/2 = 0.468774... - Vaclav Kotesovec, Jun 24 2021

A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

Original entry on oeis.org

1, 1, 2, 9, 96, 1250, 64800, 1764735, 224788480, 22499086176, 6123600000000, 408514437465750, 1308805762115174400, 133962125607455951520, 99335199198879310098432, 113040832521732593994140625, 425230288403106927476736000000, 72623663171934137824096600064000

Offset: 0

Vaclav Kotesovec, Jun 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..160

Cf. A010786, A034841, A051054, A092143, A272093.

[n eq 0 select 1 else (&*[Binomial(n,Floor(n/j)): j in [1..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024

Table[Product[Binomial[n, Floor[n/k]], {k, 1, n}], {n, 0, 20}]
Table[Product[((n + 1)/k - 1)^Floor[n/k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 24 2021 *)

[product(binomial(n,(n//j)) for j in range(1,n+1)) for n in range(31)] # G. C. Greubel, Feb 05 2024

log(a(n)) ~ n * log(n)^2 / 2. - Vaclav Kotesovec, Jun 21 2021

a(n) = Product_{k=1..n} ((n+1)/k - 1)^floor(n/k). - Vaclav Kotesovec, Jun 24 2021

cp's OEIS Frontend

A117871 Decimal expansion of Sum_{i>=1} 1/A092143(i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A092144 a(n) = A092143(n)/n!.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Extensions

A129439 An analog of Pascal's triangle: T(n,k) = A092143(n)/(A092143(n-k)*A092143(k)), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A345683 a(n) = n! * Sum_{k=1..n} 1/floor(n/k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

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