A075885 -id:A075885 - cp's OEIS frontend

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

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

A207643 a(n) = 1 + (n-1) + (n-1)[n/2-1] + (n-1)[n/2-1][n/3-1] + (n-1)[n/2-1][n/3-1][n/4-1] +... for n>0 with a(0)=1, where [x] = floor(x).

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 26, 31, 71, 129, 262, 291, 1222, 1333, 2198, 5139, 11881, 12673, 39594, 41923, 117326, 251841, 354292, 371163, 1870453, 2598577, 3456926, 7103955, 16665859, 17283113, 72923314, 75437911, 165990152, 335534913, 422310802, 695765699, 3589651696

Offset: 0

Paul D. Hanna, Feb 19 2012

Radius of convergence of g.f. A(x) is near 0.54783..., with A(1/2) = 7.2672875151872...
Compare the definition of a(n) to the trivial binomial sum:
2^(n-1) = 1 + (n-1) + (n-1)*(n/2-1) + (n-1)*(n/2-1)*(n/3-1) + (n-1)*(n/2-1)*(n/3-1)*(n/4-1) +...

			a(2) = 1 + 1 = 2; a(3) = 1 + 2 = 3;
a(4) = 1 + 3 + 3*[4/2-1] = 7;
a(5) = 1 + 4 + 4*[5/2-1] = 9;
a(6) = 1 + 5 + 5*[6/2-1] + 5*[6/2-1]*[6/3-1] = 26;
a(7) = 1 + 6 + 6*[7/2-1] + 6*[7/2-1]*[7/3-1] = 31;
a(8) = 1 + 7 + 7*[8/2-1] + 7*[8/2-1]*[8/3-1] + 7*[8/2-1]*[8/3-1]*[8/4-1] = 71; ...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A075885, A207645, A207644, A207646, A207647.

a[n_] := 1 + Sum[ Product[ Floor[(n-j)/j], {j, 1, k}], {k, 1, n/2}]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Mar 06 2013 *)

{a(n)=1+sum(k=1,n,prod(j=1,k,floor(n/j-1)))}
for(n=0,50,print1(a(n),", "))

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

a(n) = 1 + Sum_{k=1..[n/2]} Product_{j=1..k} floor( (n-j) / j ).

Equals row sums of irregular triangle A207645.

A207644 a(n) = 1 + (n-1) + (n-2)[(n-3)/2] + (n-3)[(n-4)/2][(n-5)/3] + (n-4)[(n-5)/2][(n-6)/3][(n-7)/4] +... where [x] = floor(x), with summation extending over the initial [n/2+1] products only.

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 10, 17, 30, 42, 55, 116, 172, 220, 391, 683, 1024, 1616, 2050, 3675, 6520, 9504, 12505, 22421, 35572, 56918, 85701, 138110, 202765, 326231, 503632, 860497, 1376870, 1927446, 2818531, 4892966, 7784671, 11432772, 17287295, 30423457, 46453786, 71810414

Offset: 0

Paul D. Hanna, Feb 19 2012

Radius of convergence of g.f. A(x) is near 0.637..., with A(1/phi) = 23.059250143... where phi = (sqrt(5)+1)/2.
Compare the definition of a(n) with the binomial sum:
Fibonacci(n) = 1 + (n-1) + (n-2)*((n-3)/2) + (n-3)*((n-4)/2)*((n-5)/3) + (n-4)*((n-5)/2)*((n-6)/3)*((n-7)/4) +...
with summation extending over the initial [n/2+1] products only.

			a(3) = 1 + 2 = 3;
a(4) = 1 + 3 + 2*[1/2] = 4;
a(5) = 1 + 4 + 3*[2/2] = 8;
a(6) = 1 + 5 + 4*[3/2] + 3*[2/2]*[1/3] = 10;
a(7) = 1 + 6 + 5*[4/2] + 4*[3/2]*[2/3] = 17;
a(8) = 1 + 7 + 6*[5/2] + 5*[4/2]*[3/3] + 4*[3/2]*[2/3]*[1/4] = 30;
a(9) = 1 + 8 + 7*[6/2] + 6*[5/2]*[4/3] + 5*[4/2]*[3/3]*[2/4] = 42; ...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A075885, A207643, A207645, A207646, A207647.

a[n_] := 1 + Sum[ Product[ Floor[ (n-k-j+1)/j ], {j, 1, k}], {k, 1, n/2}]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Mar 06 2013 *)

{a(n)=1+sum(k=1,n\2,prod(j=1,k,floor((n-k-j+1)/j)))}
for(n=0,60,print1(a(n),", "))

a(n) = 1 + Sum_{k=1..[n/2]} Product_{j=1..k} floor( (n-k-j+1) / j ).

Equals the antidiagonal sums of triangle A207645.

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).

A331213 a(n) = 1 + Sum_{i=1..n} (-1)^i * Product_{j=1..i} floor(n/j).

Original entry on oeis.org

1, 0, 1, -2, 5, -4, 13, -27, 89, -80, 191, -450, 2365, -1182, 3221, -13034, 40433, -22320, 96373, -193761, 772981, -728930, 1599357, -3428425, 21411337, -13595724, 31407273, -110011850, 377746853, -198079308, 1096983421, -2241234465, 7565512161, -6472208192

Offset: 0

Seiichi Manyama, Jan 12 2020

sign

Compare to the exponential series: exp(-n) = 1 - n + n*(n/2) - n*(n/2)*(n/3) + n*(n/2)*(n/3)*(n/4) - ...

			a(4) = 1 - 4 + 4*floor(4/2) - 4*floor(4/2)*floor(4/3) + 4*floor(4/2)*floor(4/3)*floor(4/4) = 1 - 4 + 4*2 - 4*2*1 + 4*2*1*1 = 5.

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

Cf. similar sequences: A075885 (b=1), A208060 (b=2).

Cf. A010786.

[1] cat [1+&+[(-1)^i*(&*[Floor(n/j):j in [1..i]]):i in [1..n]]:n in [1..33]]; // Marius A. Burtea, Jan 13 2020

a[n_] := 1 + Sum[(-1)^i * Product[Floor[n/j], {j, 1, i}],{i, 1, n}]; Array[a, 34, 0] (* Amiram Eldar, Jan 13 2020 *)

{a(n) = 1+sum(i=1, n, (-1)^i*prod(j=1, i, floor(n/j)))}

A213907 a(n) = 1 + n + n{n/2} + n{n/2}{n/3} + n{n/2}{n/3}{n/4} +... where {x} = [x+1/2] = round(x).

Original entry on oeis.org

1, 3, 9, 34, 61, 261, 709, 1324, 4937, 15040, 28561, 107262, 248341, 522445, 1972363, 7591936, 8835345, 26421129, 145475533, 183752250, 701184621, 2234736295, 2996725227, 15105451596, 32483720761, 77618520551, 217809217211, 625456400842, 1638545943301

Offset: 0

Paul D. Hanna, Jun 24 2012

nonn

			a(2) = 1 + 2 + 2*1 + 2*1*1 + 2*1*1*1 = 9.
a(3) = 1 + 3 + 3*2 + 3*2*1 + 3*2*1*1 + 3*2*1*1*1 + 3*2*1*1*1*1 = 34.

Cf. A075885.

{a(n)=1+sum(m=1, 2*n, prod(k=1, m, round(n/k)))}
for(n=0, 60, print1(a(n), ", "))

cp's OEIS Frontend

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

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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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A207643 a(n) = 1 + (n-1) + (n-1)*[n/2-1] + (n-1)*[n/2-1]*[n/3-1] + (n-1)*[n/2-1]*[n/3-1]*[n/4-1] +... for n>0 with a(0)=1, where [x] = floor(x).

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A207644 a(n) = 1 + (n-1) + (n-2)*[(n-3)/2] + (n-3)*[(n-4)/2]*[(n-5)/3] + (n-4)*[(n-5)/2]*[(n-6)/3]*[(n-7)/4] +... where [x] = floor(x), with summation extending over the initial [n/2+1] products only.

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A208060 a(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] + ... where [x]=floor(x).

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A331213 a(n) = 1 + Sum_{i=1..n} (-1)^i * Product_{j=1..i} floor(n/j).

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

A207643 a(n) = 1 + (n-1) + (n-1)[n/2-1] + (n-1)[n/2-1][n/3-1] + (n-1)[n/2-1][n/3-1][n/4-1] +... for n>0 with a(0)=1, where [x] = floor(x).

A207644 a(n) = 1 + (n-1) + (n-2)[(n-3)/2] + (n-3)[(n-4)/2][(n-5)/3] + (n-4)[(n-5)/2][(n-6)/3][(n-7)/4] +... where [x] = floor(x), with summation extending over the initial [n/2+1] products only.

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).

A213907 a(n) = 1 + n + n{n/2} + n{n/2}{n/3} + n{n/2}{n/3}{n/4} +... where {x} = [x+1/2] = round(x).