A033312 -id:A033312 - cp's OEIS frontend

A214015 Number of permutations A(n,k) in S_n with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 1, 0, 1, 1, 2, 6, 14, 1, 0, 1, 1, 2, 6, 23, 42, 1, 0, 1, 1, 2, 6, 24, 103, 132, 1, 0, 1, 1, 2, 6, 24, 119, 513, 429, 1, 0, 1, 1, 2, 6, 24, 120, 694, 2761, 1430, 1, 0, 1, 1, 2, 6, 24, 120, 719, 4582, 15767, 4862, 1, 0

Offset: 0

Alois P. Heinz, Jul 01 2012

A(n,k) is also the sum of the squares of numbers of standard Young tableaux (SYT) of height <= k over all partitions of n.
This array is a larger and reflected version of A047888.
Column k>1 is asymptotic to (Product_{j=1..k} j!) * k^(2*n + k^2/2) / (Pi^((k-1)/2) * 2^((k-1)*(k+2)/2) * n^((k^2-1)/2)). - Vaclav Kotesovec, Sep 10 2014

			A(4,2) = 14 because 14 permutations of {1,2,3,4} do not contain an increasing subsequence of length > 2: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321.  Permutation 1423 is not counted because it contains the noncontiguous increasing subsequence 123.
A(4,2) = 14 = 2^2 + 3^2 + 1^2 because the partitions of 4 with <= 2 parts are [2,2], [3,1], [4] with 2, 3, 1 standard Young tableaux, respectively:
  +------+  +------+  +---------+  +---------+  +---------+  +------------+
  | 1  3 |  | 1  2 |  | 1  3  4 |  | 1  2  4 |  | 1  2  3 |  | 1  2  3  4 |
  | 2  4 |  | 3  4 |  | 2 .-----+  | 3 .-----+  | 4 .-----+  +------------+
  +------+  +------+  +---+        +---+        +---+
Square array A(n,k) begins:
  1,  1,   1,    1,    1,    1,    1,    1, ...
  0,  1,   1,    1,    1,    1,    1,    1, ...
  0,  1,   2,    2,    2,    2,    2,    2, ...
  0,  1,   5,    6,    6,    6,    6,    6, ...
  0,  1,  14,   23,   24,   24,   24,   24, ...
  0,  1,  42,  103,  119,  120,  120,  120, ...
  0,  1, 132,  513,  694,  719,  720,  720, ...
  0,  1, 429, 2761, 4582, 5003, 5039, 5040, ...

Alois P. Heinz, Antidiagonals n = 0..70, flattened
Wikipedia, Longest increasing subsequence problem
Wikipedia, Young tableau

Columns k=0-10 give: A000007, A000012, A000108, A005802, A047889, A047890, A052399, A072131, A072132, A072133, A072167.
Differences between A000142 and columns k=0-9 give: A000142 (for n>0), A033312, A056986, A158005, A158432, A159139, A159175, A217675, A217676, A217677.
Main diagonal and first lower diagonal give: A000142, A033312.
A(2n,n-1) gives A269042(n) for n>0.
Cf. A047887, A047888, A182172, A208447, A214152, A267479.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
A:= (n, k)-> `if`(k>=n, n!, g(n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]! / Product[Product[1+l[[i]]-j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
A[n_, k_] := If[k >= n, n!, g[n, k, {}]];
Table [Table [A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.

Original entry on oeis.org

1, 1, 4, 52, 2776, 695541, 927908528, 7303437156115, 371421772559819369, 132348505150329265211927, 355539706668772869353964510735, 7698296698535929906799439134946965681, 1428662247641961794158621629098030994429958386, 2405509035205023556420199819453960482395657232596725626

Offset: 0

Antti Karttunen, Aug 10 2018

nonn

Number of factorizations of the superprimorial A006939(n) into factors > 1. - Gus Wiseman, Aug 21 2020

			For n = 2 we have a multiset {1, 2, 2} which can be partitioned as {{1}, {2}, {2}} or {{1, 2}, {2}} or {{1}, {2, 2}} or {{1, 2, 2}}, thus a(2) = 4.

Index entries for sequences related to partitions

Cf. A033312, A317826.
Subsequence of A317828.
A000142 counts submultisets of the same multiset.
A022915 counts permutations of the same multiset.
A337069 is the strict case.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A076716 counts factorizations of factorials.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A181818 lists products of superprimorials, with complement A336426.
Cf. A000178, A002110, A022559, A027423, A124010, A303279, A318284, A322583, A336417, A336496.

g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> g(mul(ithprime(i)^i, i=1..n)$2):
seq(a(n), n=0..5);  # Alois P. Heinz, Jul 26 2020

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[chern[n]]],{n,3}] (* Gus Wiseman, Aug 21 2020 *)

\\ See A318284 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

a(n) = A317826(A033312(n+1)) = A317826((n+1)!-1) = A001055(A076954(n)).
a(n) = A001055(A006939(n)). - Gus Wiseman, Aug 21 2020
a(n) = A318284(A002110(n)). - Andrew Howroyd, Aug 31 2020

a(0)=1 prepended and a(7) added by Alois P. Heinz, Jul 26 2020

a(8)-a(13) from Andrew Howroyd, Aug 31 2020

A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 5, 1, 0, 0, 1, 43, 23, 1, 0, 0, 1, 374, 1879, 119, 1, 0, 0, 1, 3199, 173891, 102011, 719, 1, 0, 0, 1, 26945, 16140983, 117392909, 7235651, 5039, 1, 0, 0, 1, 224296, 1474050783, 142951955371, 117108036719, 674641325, 40319, 1

Offset: 0

Alois P. Heinz, Feb 19 2016

			Square array A(n,k) begins:
  0,   0,      0,         0,            0,               0, ...
  1,   0,      0,         0,            0,               0, ...
  1,   1,      1,         1,            1,               1, ...
  1,   5,     43,       374,         3199,           26945, ...
  1,  23,   1879,    173891,     16140983,      1474050783, ...
  1, 119, 102011, 117392909, 142951955371, 173996758190594, ...

Alois P. Heinz, Antidiagonals n = 0..50, flattened
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Columns k=0-10 give: A057427, A033312, A267532, A269113, A269114, A269115, A269116, A269117, A269118, A269119, A269120.
Rows n=0-10 give: A000004, A000007, A000012, A269121, A269122, A269123, A269124, A269125, A269126, A269127, A269128.
Main diagonal gives: A268751.
Cf. A047909, A089759, A187783, A331562.

g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
      binomial(n-1, l[-1]-1)+add(f(sort(subsop(j=l[j]
      -1, l))), j=1..nops(l)-1))(add(i, i=l))
    end:
f:= l->(n->`if`(n=0, 1, `if`(l[1]=0, 0, `if`(n=1 or l[-1]=1, 1,
    `if`(n=2, binomial(l[1]+l[2], l[1])-1, g(l))))))(nops(l)):
A:= (n, k)-> (k*n)!/k!^n - f([k$n]):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(k, p, j, l, t) option remember;
      `if`(k=0, (-1)^t/l!, `if`(p<0, 0, add(b(k-i, p-1,
       j+1, l+i*j, irem(t+i*j, 2))/(i!*p!^i), i=0..k)))
    end:
A:= (n, k)-> (n*k)!*(1/k!^n-b(n, k-1, 1, 0, irem(n, 2))*n!):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 03 2016

b[k_, p_, j_, l_, t_] := b[k, p, j, l, t] = If[k == 0, (-1)^t/l!, If[p < 0, 0, Sum[b[k-i, p-1, j+1, l + i j, Mod[t + i j, 2]]/(i! p!^i), {i, 0, k}]] ];
A[n_, k_] := (n k)! (1/k!^n - b[n, k-1, 1, 0, Mod[n, 2]] n!); Table[ Table[ A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

A(n,k) = A089759(k,n) - A047909(k,n) = A187783(n,k) - A047909(k,n).

A125714 Alfred Moessner's factorial triangle.

Original entry on oeis.org

1, 2, 3, 6, 11, 6, 24, 50, 35, 10, 120, 274, 225, 85, 15, 720, 1764, 1624, 735, 175, 21, 5040, 13068, 13132, 6769, 1960, 322, 28, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 3628800

Offset: 1

Gary W. Adamson, Dec 01 2006

Successive numbers arising from the Moessner construction of the factorial numbers. - N. J. A. Sloane, Jul 27 2021
Row sums of the triangle = 1, 5, 23, 119, 719, ...(matching the terms 0, 0, 1, 5, 23, 119, 719, ...; of A033312).
The name of the triangle derives from the fact that A125714(A000124(n)) = A000142(n) for n > 0. Moessner's method uses only additions to compute the factorial n!. - Peter Luschny, Jan 27 2009
If n = (m^2+m+2)/2 then a(n) = (m+1)!. For example, taking m = 3, n = 7, and indeed a(7) = 4! = 24. - N. J. A. Sloane, Jul 27 2021

			An "x" prefaced before each term will indicate the term following the x being circled.
x1 2 x3 4 5 x6 7 8 9 x10 11 12 13 14 x15 ...
__x2 6 x11 18 26 x35 46 58 71 x85 ...
_____________x6 24 x50 96 154 x225 ...
_________________________x24 120 x274 ...
___________________________________________x120 ...
...
i.e., circle the triangular terms in row 1. In row 2, take partial sums of the uncircled terms and circle the terms offset one place to the left of the triangular terms in row 1. Continue in subsequent rows with analogous operations. The triangle consists of the infinite set of terms prefaced with the x (circled on page 64 of "The Book of Numbers").

J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996. Sequence can be seen by reading the successive circled numbers in the "factorial" section on page 64 (based on the work of Alfred Moessner).

Joshua Zucker, Table of n, a(n) for n = 1..66
G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
R. Krebbers, L. Parlant, and A. Silva, Moessner's theorem: an exercise in coinductive reasoning in Coq, Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
Ivan Paasche, Ein neuer Beweis des Moessnerschen Satzes S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
Ivan Paasche, Beweis des Moessnerschen Satzes mittels linearer Transformationen, Arch. Math. (Basel) 6 (1955), 194-199.
Ivan Paasche, Eine Verallgemeinerung des Moessnerschen Satzes, Compositio Math. 12 (1956), 263-270.
Hans Salié, Bemerkung zu einem Satz von A. Moessner, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Cf. A000217, A003056, A008275, A033312, A346004-A346007, A346595.

a := proc(n) local s,m,k,i; s := array(0..n); s[0] := 1;
for m from 1 to n do s[m] := 0; for k from m by - 1 to 1 do
for i from 1 to k do s[i] := s[i] + s[i - 1] od; lprint(s[k]);
if k = n then return(s[n]) fi od; lprint("-") od end: # Peter Luschny, Jan 27 2009
with(combinat);
s:=stirling1;
A003056 := proc(n) floor((sqrt(1+8*n)-1)/2) ; end proc:
T:=n->n*(n+1)/2; # A000217
g:=proc(n) local i,j,t; global T,A003056;
j:=A003056(n-1)+1;
t:=T(j);
for i from 0 to t-1 do
if ((n+i) mod t) = 0 then return(abs(s(j+1,j-i))); fi;
od;
end;
[seq(g(n),n=1..80)]; # N. J. A. Sloane, Jul 27 2021

n = 10; A125714 = Reap[ ClearAll[s]; s[0] = 1; For[m = 1, m <= n, m++, s[m] = 0; For[k = m, k >= 1, k--, For[i = 1, i <= k, i++, s[i] = s[i] + s[i-1]]; Sow[s[k]]; If[k == n, Print[n, "! = ", s[n]]; Break[]]]]][[2, 1]] (* Jean-François Alcover, Jun 29 2012, after Peter Luschny *)

T(n, k)={ my( s=vector(n)); for( m=1, n, forstep( j=m,1,-1, s[1]++; for( i=2, j, s[i] += s[i-1]));
k<0 && print(vecextract(s,Str(m"..1"))));
if( k>0,s[n+1-k],vecextract(s,"-1..1"))} /* returns T[n,k], or the whole n-th row if k is not given, prints row 1...n of the triangle if k<0 */ \\ M. F. Hasler, Dec 03 2010

Starting with the natural numbers, circle each triangular number. Underneath, take partial sums of the uncircled terms and circle the terms in this row which are offset one place to the left of the circled 1, 3, 6, 10, ... in the first row. Repeat with analogous operations in succeeding rows. The circled terms in the infinite set become the triangle.

Given n, let j = A003056(n-1)+1 and set t = j*(j+1)/2. Then, for 0 <= i < t, if n == -i (mod t), a(n) = abs(Stirling_1(j+1,j-i)). - N. J. A. Sloane, Jul 27 2021

More terms from Joshua Zucker, Jun 17 2007

A005096 a(n) = n! - n.

Original entry on oeis.org

1, 0, 0, 3, 20, 115, 714, 5033, 40312, 362871, 3628790, 39916789, 479001588, 6227020787, 87178291186, 1307674367985, 20922789887984, 355687428095983, 6402373705727982, 121645100408831981, 2432902008176639980, 51090942171709439979, 1124000727777607679978

Offset: 0

N. J. A. Sloane

McCombinations: in 2002, McDonald's advertised a McChoice menu of 8 items under the heading "40,312 combinations" rather than the more obvious 2^8-1=255 (A000225). The Advertising Standards Authority "considered that the number quoted in the advertisement was not necessarily so exaggerated as to be misleading". - Henry Bottomley, May 01 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Advertising Standards Authority, McDonald's Restaurants Ltd (archived May 10 2004).
Index entries for sequences related to factorial numbers

Cf. A000142, A033312.

[Factorial(n) - n: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

A005096:=n->n!-n: seq(A005096(n), n=0..25); # Wesley Ivan Hurt, Oct 28 2014

lst={};Do[AppendTo[lst, n!-n], {n, 3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 20 2008 *)
Table[n! - n, {n, 0, 25}] (* Wesley Ivan Hurt, Oct 28 2014 *)

a(n)=n!-n \\ Charles R Greathouse IV, Oct 28 2014

a(n) = n*a(n-1)+n(n-2) = n*A033312(n-1) = A000142(n)-n. - Henry Bottomley, May 01 2003

E.g.f.: 1/(1 - x) - x*exp(x). - Ilya Gutkovskiy, Jan 27 2017

A214152 Number of permutations T(n,k) in S_n containing an increasing subsequence of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 24, 23, 10, 1, 120, 119, 78, 17, 1, 720, 719, 588, 207, 26, 1, 5040, 5039, 4611, 2279, 458, 37, 1, 40320, 40319, 38890, 24553, 6996, 891, 50, 1, 362880, 362879, 358018, 268521, 101072, 18043, 1578, 65, 1, 3628800, 3628799, 3612004, 3042210, 1438112, 337210, 40884, 2603, 82, 1

Offset: 1

Alois P. Heinz, Jul 05 2012

			T(3,2) = 5.  All 3! = 6 permutations of {1,2,3} contain an increasing subsequence of length 2 with the exception of 321.
Triangle T(n,k) begins:
     1;
     2,    1;
     6,    5,    1;
    24,   23,   10,    1;
   120,  119,   78,   17,   1;
   720,  719,  588,  207,  26,  1;
  5040, 5039, 4611, 2279, 458, 37,  1;
  ...

Alois P. Heinz, Rows n = 1..55, flattened
Eric Weisstein's World of Mathematics, Permutation Pattern
Wikipedia, Longest increasing subsequence problem
Wikipedia, Young tableau

Columns k=1-10 give: A000142 (for n>0), A033312, A056986, A158005, A158432, A159139, A159175, A217675, A217676, A217677.
Row sums give: A003316.
T(2n,n) gives A269021.
Diagonal and lower diagonals give: A000012, A002522, A217200, A217193.
Cf. A047874, A214015.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= (n, k)-> n! -g(n, k-1, []):
seq(seq(T(n, k), k=1..n), n=1..12);

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]! / Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] ]; g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1&, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; t[n_, k_] := n! - g[n, k-1, {}]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

T(n,k) = Sum_{i=k..n} A047874(n,i).

T(n,k) = A000142(n) - A214015(n,k-1).

A054991 Number of prime divisors of n! - 1 (counted with multiplicity).

Original entry on oeis.org

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

Offset: 1

Arne Ring (arne.ring(AT)epost.de), May 30 2000

nonn

The series is related to the product of primes and the "proof" of the existence of infinite many prime twins.

			a(2)=0 because 2! - 1 = 1 (and this is not a prime number) a(5)=2 because 5! -1 = 119 = 7 * 17

Amiram Eldar, Table of n, a(n) for n = 1..135
R. G. Wilson v, Explicit factorizations
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences

Cf. A033312, A054988, A054989, A054990, A054992.

a[q_] := Module[{x, n}, x=FactorInteger[q!-1]; n=Length[x]; Sum[Table[x[[i]][[2]], {i, n}][[j]], {j, n}]]
A054991[n_Integer] := PrimeOmega[n! - 1]; A054991[1] = 0; Table[A054991[n], {n,2,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2011 *)

More terms from Robert G. Wilson v, Mar 24 2001

More terms from Amiram Eldar, Oct 03 2019

A002582 Largest prime factor of n! - 1.

Original entry on oeis.org

1, 5, 23, 17, 719, 5039, 1753, 2999, 125131, 7853, 479001599, 3593203, 87178291199, 1510259, 6880233439, 256443711677, 478749547, 78143369, 19499250680671, 4826713612027, 170006681813, 498390560021687969, 991459181683, 114776274341482621993

Offset: 2

N. J. A. Sloane

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Table of n, a(n) for n = 2..135
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations
J. W. Wrench, Jr., Letters to N. J. A. Sloane, Feb 1974

Cf. A002583, A033312, A054415, A056110.

[1] cat [Maximum(PrimeDivisors(Factorial(n)-1)): n in [3..30]]; // Vincenzo Librandi, Feb 14 2020

Table[FactorInteger[n! - 1][[-1, 1]], {n, 2, 25}] (* Harvey P. Dale, Aug 29 2011 *)

a(n)=if(n>2,my(f=factor(n!-1)[,1]);f[#f],1) \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n and lim sup a(n)/n > 2. - Charles R Greathouse IV, Dec 05 2012

More terms from Robert G. Wilson v, Aug 01 2000

A028418 Sum over all n! permutations of n letters of maximum cycle length.

Original entry on oeis.org

1, 3, 13, 67, 411, 2911, 23563, 213543, 2149927, 23759791, 286370151, 3734929903, 52455166063, 788704078527, 12648867695311, 215433088624351, 3884791172487903, 73919882720901823, 1480542628345939807, 31128584449987511871, 685635398619169059391

Offset: 1

Joe Keane (jgk(AT)jgk.org)

nonn

Sum the n-permutations having at least 1 cycle of length >= i for all i >= 1. A000142 + A033312 + A066052 + A202364 + ... The summation is precisely that indicated in the title since each permutation whose longest cycle = i is counted i times. - Geoffrey Critzer, Jan 09 2013

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 183.
R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 358.

Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 142 terms from Thomas Dybdahl Ahle)
Ph. Flajolet and A. Odlyzko, Singularity analysis of generating functions, p. 22.

Cf. A006128, A028417, A060014, A126074.

Column k=1 of A322384.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, max(m,j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..25);  # Alois P. Heinz, May 14 2016

kmax = 19; gf[x_] = Sum[ 1/(1-x) - 1/(E^((x^(1+k)*Hypergeometric2F1[1+k, 1, 2+k, x])/ (1+k))*(1-x)), {k, 0, kmax}];
a[n_] := n!*Coefficient[Series[gf[x], {x, 0, kmax}], x^n]; Array[a, kmax]
(* Jean-François Alcover, Jun 22 2011, after e.g.f. *)
a[ n_] := If[ n < 1, 0, 1 + Total @ Apply[ Max, Map[ Length, First /@ PermutationCycles /@ Drop[ Permutations @ Range @ n, 1], {2}], 1]]; (* Michael Somos, Aug 19 2018 *)

E.g.f.: Sum_{k>=0} (1/(1-x) - exp(Sum_{j=1..k} x^j/j)).
a(n) = f(n, 0, n, n!) where f(L, r, n, m) = m*r if r >= l, otherwise Sum_{k=0..L-1} (f(k, max(L-k,r), n-1, m/n) + (n-L)*f(L, r, n-1, m/n)). - Thomas Dybdahl Ahle, Aug 15 2011
a(n) = Sum_{k=1..n} k * A126074(n,k). - Alois P. Heinz, May 17 2016

More terms from Vladeta Jovovic, Sep 19 2002

More terms from Thomas Dybdahl Ahle, Aug 15 2011

A139174 a(n) = (n!-4)/4.

Original entry on oeis.org

5, 29, 179, 1259, 10079, 90719, 907199, 9979199, 119750399, 1556755199, 21794572799, 326918591999, 5230697471999, 88921857023999, 1600593426431999, 30411275102207999, 608225502044159999

Offset: 4

Artur Jasinski, Apr 11 2008

In the solution set of the equation, n! + s^2 = m^2, for any n>=4 the maximum value of s is (n!-4)/4. - Sudipta Mallick, Jul 27 2019

Sudipta Mallick, Table of n, a(n) for n = 4..400 (terms 4..300 from Vincenzo Librandi)

Cf. A033312, A139172, A139173, A139174, A139175, A139176, A139177, A139183, A139184.

[(Factorial(n)-4)/4: n in [4..25]]; // Vincenzo Librandi, Jul 20 2011

Mathematica
```
Table[(n! - 4)/4, {n, 4, 20}]
```

a(n) = numerator(((n+2)!-4)/(7*(n+2)!)), with offset 2. [Gary Detlefs, Nov 07 2010]

E.g.f.: -exp(x)-(12+x^2*(-6+(-4+x)*x))/(12*(-1+x)). - Stefano Spezia, Sep 08 2019

Definition corrected by Gary Detlefs, Nov 07 2010

cp's OEIS Frontend

A214015 Number of permutations A(n,k) in S_n with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A317829 Number of set partitions of multiset {1, 2, 2, 3, 3, 3, ..., n X n}.

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A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A125714 Alfred Moessner's factorial triangle.

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A005096 a(n) = n! - n.

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A214152 Number of permutations T(n,k) in S_n containing an increasing subsequence of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A054991 Number of prime divisors of n! - 1 (counted with multiplicity).

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