A000793 -id:A000793 - cp's OEIS frontend

A009490 Number of distinct orders of permutations of n objects; number of nonisomorphic cyclic subgroups of symmetric group S_n.

1, 1, 2, 3, 4, 6, 6, 9, 11, 14, 16, 20, 23, 27, 31, 35, 43, 47, 55, 61, 70, 78, 88, 98, 111, 123, 136, 152, 168, 187, 204, 225, 248, 271, 296, 325, 356, 387, 418, 455, 495, 537, 581, 629, 678, 732, 787, 851, 918, 986, 1056, 1133, 1217, 1307, 1399, 1498, 1600, 1708, 1823

Offset: 0

David W. Wilson

Also number of different LCM's of partitions of n.

a(n) <= A023893(n), which counts the nonisomorphic Abelian subgroups of S_n. - M. F. Hasler, May 24 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
L. Elliott, A. Levine, and J. D. Mitchell, Counting monogenic monoids and inverse monoids, arXiv:2303.12387 [math.GR], 2023.
Index entries for sequences related to lcm's

Cf. A051613 (first differences), A000792, A000793, A034891, A051625 (all cyclic subgroups), A256067.

b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, b(n, i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n)))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

Table[ Length[ Union[ Apply[ LCM, Partitions[ n ], 1 ] ] ], {n, 30} ]
f[n_] := Length@ Union[LCM @@@ IntegerPartitions@ n]; Array[f, 60, 0]
(* Caution, the following is Extremely Slow and Resource Intensive *) CoefficientList[ Series[ Expand[ Product[1 + Sum[x^(Prime@ i^k), {k, 4}], {i, 10}]/(1 - x)], {x, 0, 30}], x]
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0 || i<1, 1, b[n, i-1]+Sum[b[n-p^j, i-1], {j, 1, Log[p, n]}]]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)

/* compute David W. Wilson's g.f., needs <1 sec for 1000 terms */
N=1000;  x='x+O('x^N); /* N terms */
gf=1; /* generating function */
{ forprime(p=2,N,
    sm = 1;  pp=p;  /* sum;  prime power */
    while ( ppJoerg Arndt, Jan 19 2011 */

a(n) = Sum_{k=0..n} b(k), where b(k) is the number of partitions of k into distinct prime power parts (1 excluded) (A051613). - Vladeta Jovovic

G.f.: (Product_{p prime} (1 + Sum_{k >= 1} x^(p^k))) / (1-x). - David W. Wilson, Apr 19 2000

A222029 Triangle of number of functions in a size n set for which the sequence of composition powers ends in a length k cycle.

Original entry on oeis.org

1, 1, 3, 1, 16, 9, 2, 125, 93, 32, 6, 1296, 1155, 480, 150, 24, 20, 16807, 17025, 7880, 3240, 864, 840, 262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420, 4782969, 5752131, 3009888, 1692180, 653184, 773920, 46080, 5040, 0, 32256, 0, 26880, 0, 0, 2688

Offset: 0

Chad Brewbaker, May 14 2013

If you take the powers of a finite function you generate a lollipop graph. This table organizes the lollipops by cycle size. The table organized by total lollipop size with the tail included is A225725.

Warning: For T(n,k) after the sixth row there are zero entries and k can be greater than n: T(7,k) = |{1=>262144, 2=>292383, 3=>145320, 4=>71610, 5=>24192, 6=>26250, 7=>720, 8=>0, 9=>0, 10=>504, 11=>0, 12=>420}|.

			T(1,1) = |{[0]}|, T(2,1) = |{[0,0],[0,1],[1,1]}|, T(2,2) = |{[0,1]}|.
Triangle starts:
       1;
       1;
       3,      1;
      16,      9,      2;
     125,     93,     32,     6;
    1296,   1155,    480,   150,    24,    20;
   16807,  17025,   7880,  3240,   864,   840;
  262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420;
  ...

Alois P. Heinz, Rows n = 0..30, flattened
Chad Brewbaker, Ruby program for A222029

Columns k=1-10 give A000272, A163951, A163952, A291110, A291111, A291112, A291113, A291114, A291115, A291116.
Rows sums give A000312.
Row lengths are A000793.
Number of nonzero elements of rows give A009490.
Last elements of rows give A162682.
Main diagonal gives A290961.
Cf. A057731 (the same for permutations), A290932.

b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
      b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(
         b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 14 2017

b[n_, m_]:=b[n, m]=If[n==0, x^m, Sum[(j - 1)!*b[n - j, LCM[m, j]] Binomial[n - 1, j - 1], {j, n}]]; T[n_]:=If[n==0, {1}, Drop[CoefficientList[Sum[b[j, 1]n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], x], 1]]; Table[T[n], {n, 0, 10}]//Flatten (* Indranil Ghosh, Aug 17 2017 *)

from sympy.core.cache import cacheit
from sympy import binomial, Symbol, lcm, factorial as f, Poly, flatten
x=Symbol('x')
@cacheit
def b(n, m): return x**m if n==0 else sum([f(j - 1)*b(n - j, lcm(m, j))*binomial(n - 1, j - 1) for j in range(1, n + 1)])
def T(n): return Poly(sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)]),x).all_coeffs()[::-1][1:]
print([T(n) for n in range(11)]) # Indranil Ghosh, Aug 17 2017

Sum_{k=1..A000793(n)} k * T(n,k) = A290932. - Alois P. Heinz, Aug 14 2017

T(0,1)=1 prepended by Alois P. Heinz, Aug 14 2017

A060014 Sum of orders of all permutations of n letters.

Original entry on oeis.org

1, 1, 3, 13, 67, 471, 3271, 31333, 299223, 3291487, 39020911, 543960561, 7466726983, 118551513523, 1917378505407, 32405299019941, 608246253790591, 12219834139189263, 253767339725277823, 5591088918313739017, 126036990829657056711, 2956563745611392385211

Offset: 0

N. J. A. Sloane, Mar 17 2001

Conjecture: This sequence eventually becomes cyclic mod n for all n. - Isaac Saffold, Dec 01 2019

			For n = 4 there is 1 permutation of order 1, 9 permutations of order 2, 8 of order 3 and 6 of order 4, for a total of 67.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.2, p. 460.

Alois P. Heinz, Table of n, a(n) for n = 0..170
FindStat - Combinatorial Statistic Finder, The order of a permutation
Joshua Harrington, Lenny Jones, and Alicia Lamarche, Characterizing Finite Groups Using the Sum of the Orders of the Elements, International Journal of Combinatorics, Volume 2014, Article ID 835125, 8 pages.

Cf. A000793, A028418, A060015, A057731, A074859, A290932, A346066.

b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!
      *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 11 2017

CoefficientList[Series[Sum[n Fold[#1+MoebiusMu[n/#2] Apply[Times, Exp[x^#/#]&/@Divisors[#2] ]&,0,Divisors[n]],{n,Max[Apply[LCM,Partitions[19],1]]}],{x,0,19}],x] Range[0,19]! (* Wouter Meeussen, Jun 16 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], 1 + Total @ Apply[LCM, Map[Length, First /@ PermutationCycles /@ Drop[Permutations @ Range @ n, 1], {2}], 1]]; (* Michael Somos, Aug 19 2018 *)

\\ Naive method -- sum over cycles directly
cycleDecomposition(v:vec)={
    my(cyc=List(), flag=#v+1, n);
    while((n=vecmin(v))<#v,
        my(cur=List(), i, tmp);
        while(v[i++]!=n,);
        while(v[i] != flag,
            listput(cur, tmp=v[i]);
            v[i]=flag;
            i=tmp
        );
        if(#cur>1, listput(cyc, Vec(cur)))    \\ Omit length-1 cycles
    );
    Vec(cyc)
};
permutationOrder(v:vec)={
    lcm(apply(length, cycleDecomposition(v)))
};
a(n)=sum(i=0,n!-1,permutationOrder(numtoperm(n,i)))
\\ Charles R Greathouse IV, Nov 06 2014

A060014(n) =
{
  my(factn = n!, part, nb, i, j, res = 0);
  forpart(part = n,
    nb = 1; j = 1;
    for(i = 1, #part,
      if (i == #part || part[i + 1] != part[i],
        nb *= (i + 1 - j)! * part[i]^(i + 1 - j);
        j = i + 1));
    res += (factn / nb) * lcm(Vec(part)));
  res;
} \\ Jerome Raulin, Jul 11 2017 (much faster, O(A000041(n)) vs O(n!))

E.g.f.: Sum_{n>0} (n*Sum_{i|n} (moebius(n/i)*Product_{j|i} exp(x^j/j))). - Vladeta Jovovic, Dec 29 2004; The sum over n should run to at least A000793(k) for producing the k-th entry. - Wouter Meeussen, Jun 16 2012

a(n) = Sum_{k>=1} k* A057731(n,k). - R. J. Mathar, Aug 31 2017

More terms from Vladeta Jovovic, Mar 18 2001

More terms from Alois P. Heinz, Feb 14 2013

A225632 Irregular table read by rows: n-th row gives distinct values of successively iterated Landau-like functions for n, starting with the initial value 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 12, 1, 6, 30, 60, 1, 6, 30, 60, 1, 12, 84, 420, 1, 15, 120, 840, 1, 20, 180, 1260, 2520, 1, 30, 210, 840, 2520, 1, 30, 420, 4620, 13860, 27720, 1, 60, 660, 4620, 13860, 27720, 1, 60, 780, 8580, 60060, 180180, 360360

Offset: 1

Antti Karttunen, May 13 2013

The leftmost column of table (the initial term of each row, T(n,1)) is 1, corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition of n, after which, on the same row, each further term T(n,i) is computed by finding such a partition [p1,p2,...,pk] of n so that value of lcm(T(n,i-1), p1,p2,...,pk) is maximized, until finally A003418(n) is reached, which will be listed as the last term of row n (as the result would not change after that, if we continued the same process).

			The first fifteen rows of table are:
  1;
  1,   2;
  1,   3,    6;
  1,   4,   12;
  1,   6,   30,    60;
  1,   6,   30,    60;
  1,  12,   84,   420;
  1,  15,  120,   840;
  1,  20,  180,  1260,   2520;
  1,  30,  210,   840,   2520;
  1,  30,  420,  4620,  13860,  27720;
  1,  60,  660,  4620,  13860,  27720;
  1,  60,  780,  8580,  60060, 180180, 360360;
  1,  84, 1260, 16380, 180180, 360360;
  1, 105, 4620, 60060, 180180, 360360;

Alois P. Heinz, Rows n = 1..150, flattened
Index entries for sequences related to lcm's

Cf. A225634 (length of n-th row), A000793 (n>=2 gives the second column).
Cf. A225629 (second largest/rightmost term of n-th row).
Cf. A003418 (largest/rightmost term of n-th row).
Cf. A225630, A225631, A225635, A212721.
Cf. A225642 (row n starts from n instead of 1).
Cf. A226055 (the first term common with A225642 on the n-th row).
Cf. A225638 (distance to that first common term from the beginning of the row n).
Cf. A226056 (number of trailing terms common with A225642 on the n-th row).

b:= proc(n, i) option remember; `if`(n=0, {1},
      `if`(i<1, {}, {seq(map(x->ilcm(x, `if`(j=0, 1, i)),
       b(n-i*j, i-1))[], j=0..n/i)}))
    end:
T:= proc(n) option remember; local d, h, l, ll;
      l:= b(n$2); ll:= NULL; d:=1; h:=0;
      while d<>h do ll:= ll, d; h:= d;
        d:= max(seq(ilcm(h, i), i=l))
      od; ll
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, May 29 2013

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Table[Map[Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; T[n_] := T[n] = Module[{d, h, l, ll}, l=b[n, n]; ll={}; d=1; h=0; While[d != h, AppendTo[ll, d]; h=d; d = Max[ Table[LCM[h, i], {i, l}]]]; ll]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

A319055 Maximum product of an integer partition of n with relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 12, 18, 24, 36, 54, 72, 108, 162, 216, 324, 486, 648, 972, 1458, 1944, 2916, 4374, 5832, 8748, 13122, 17496, 26244, 39366, 52488, 78732, 118098, 157464, 236196, 354294, 472392, 708588, 1062882, 1417176, 2125764, 3188646, 4251528, 6377292

Offset: 1

Gus Wiseman, Sep 09 2018

nonn

After a(7), this appears to be the same as A319054.

Cf. A000740, A000792, A000793, A000837, A001414, A007916, A100953, A281116, A289508, A289509, A296302, A319054, A319057.

Table[Max[Times@@@Select[IntegerPartitions[n],GCD@@#==1&]],{n,20}]

A057740 Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 3, 8, 1, 15, 20, 0, 24, 1, 45, 80, 90, 144, 1, 105, 350, 630, 504, 210, 720, 1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688, 1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192

Offset: 1

Roger Cuculière, Oct 29 2000

			Triangle begins:
  1;
  1;
  1,    0,    2;
  1,    3,    8;
  1,   15,   20,     0,   24;
  1,   45,   80,    90,  144;
  1,  105,  350,   630,  504,   210,   720;
  1,  315, 1232,  3780, 1344,  5040,  5760, 0,     0,    0, 0,     0, 0, 0,  2688;
  1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192;
...

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).
Index entries for sequences related to groups

Cf. A057731 (S_n), A054522, A057741, A051593, A000793.
Column 2, 3, 4 are A048099, A001471, A051695.
See also A061129, A061130, A061131, A061132, A061133, A061134, A061135, A061136, A061137, A061138, A061139, A061140.

Magma
```
{* Order(g) : g in Alt(6) *};
```

row[n_] := (orders = PermutationOrder /@ GroupElements[AlternatingGroup[n] ]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)

More terms from N. J. A. Sloane, Nov 01 2000

Missing zero in the row for A_9 inserted by N. J. A. Sloane, Mar 27 2015

A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 18 2015

			The 5 partitions of n=4 are 1+1+1+1 (lcm=1), 1+1+2 (lcm=2), 2+2 (lcm=2), 1+3 (lcm=3) and 4 (lcm=4). So k=1, 3 and 4 appear once, k=2 appears twice.
The triangle starts:
  1 ;
  1 ;
  1  1;
  1  1  1;
  1  2  1  1;
  1  2  1  1  1  1;
  1  3  2  2  1  2;
  1  3  2  2  1  3  1  0  0  1  0  1;
  ...

Alois P. Heinz, Rows n = 0..30, flattened

Cf. A000041 (row sums), A000793 (row lengths), A213952, A074761 (diagonal), A074752 (6th column), A008642 (4th column), A002266 (5th column), A002264 (3rd column), A132270 (7th column), A008643 (8th column), A008649 (9th column), A258470 (10th column).

Cf. A009490 (number of nonzero terms of rows), A074064 (last elements of rows), A168532 (the same for gcd), A181844 (Sum k*T(n,k)).

A256067 := proc(n,k)
        local a,p ;
        a := 0 ;
        for p in combinat[partition](n) do
                ilcm(op(p)) ;
                if % = k then
                        a := a+1 ;
                end if;
        end do:
        a;
end proc:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 27 2015

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i-1] + Function[{p}, Sum[ Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n-i*j, i-1], {j, 1, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

T(0,1)=1 prepended by Alois P. Heinz, Mar 27 2015

A074064 Number of cycle types of degree-n permutations having the maximum possible order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4

Offset: 0

Vladeta Jovovic, Sep 15 2002

			For n = 22 we have 4 such cycle types: [1, 1, 1, 3, 4, 5, 7], [1, 2, 3, 4, 5, 7], [3, 3, 4, 5, 7], [4, 5, 6, 7].

Alois P. Heinz, Table of n, a(n) for n = 0..180

Cf. A000793, A074859, A256067, A256554.

A000793 := proc(n) option remember; local l,p,i ; l := 1: p := combinat[partition](n): for i from 1 to combinat[numbpart](n) do if ilcm( p[i][j] $ j=1..nops(p[i])) > l then l := ilcm( p[i][j] $ j=1..nops(p[i])) ; fi: od: RETURN(l) ; end proc:
taylInv := proc(i,n) local resul,j,idiv,k ; resul := 1 ; idiv := numtheory[divisors](i) ; for k from 1 to nops(idiv) do j := op(k,idiv) ; resul := resul*taylor(1/(1-x^j),x=0,n+1) ; resul := convert(taylor(resul,x=0,n+1),polynom) ; od ; coeftayl(resul,x=0,n) ; end proc:
A074064 := proc(n) local resul,a793,dvs,i,k ; resul := 0: a793 := A000793(n) ; dvs := numtheory[divisors](a793) ; for k from 1 to nops(dvs) do i := op(k,dvs) ; resul := resul+numtheory[mobius](a793/i)*taylInv(i,n) ; od : RETURN(resul) ; end proc: # R. J. Mathar, Mar 30 2007

b[n_, i_] := b[n, i] = Module[{p}, p = If[i < 1, 1, Prime[i]]; If[n == 0 || i < 1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor}]]]];
g[n_] := g[n] = b[n, If[n < 8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];
a[n_] := a[n] = SeriesCoefficient[Sum[MoebiusMu[g[n]/i]/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[g[n]]}] + O[x]^(n+1), n];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 25 2017, after Alois P. Heinz *)

Coefficient of x^n in expansion of Sum_{i divides A000793(n)} mu(A000793(n)/i)*1/Product_{j divides i} (1-x^j).

More terms from R. J. Mathar, Mar 30 2007
More terms from Sean A. Irvine, Oct 04 2011
More terms from Alois P. Heinz, Mar 29 2015

A051593 Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 5, 7, 15, 15, 21, 21, 35, 35, 60, 105, 105, 105, 140, 210, 210, 420, 420, 420, 420, 840, 1155, 1260, 1365, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 15015, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060

Offset: 0

Vladeta Jovovic

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Index entries for sequences related to groups

Cf. A057742, A057743, A057740, A000793.

(* a3 = A000793  a4 = A051704 *) a3[n_] := Max[LCM @@@ IntegerPartitions[n]]; a4[n_] := (pp = Reap[ Do[ pk = p^k; If[pk <= n, Sow[pk]], {p, Prime[ Range[2, PrimePi[n]]]}, {k, 1, Ceiling[ Log[3, n]]}]][[2, 1]]; sel = Select[ IntegerPartitions[n, All, pp], Length[#] == Length[ Union[#] && !MatchQ[#, {_, x_, _, y_, _} /; GCD[x, y] != 1]] &]; Max[Times @@@ sel]); a4[0] = 1; a4[1] = a4[2] = a4[4] = a4[6] = 0; a[n_] := Max[a3[n - 2], a4[n - 1], a4[n]]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Sep 11 2012, from formula *)

a(n)={my(m=1); forpart(p=n, if(sum(i=1, #p, p[i]-1)%2==0, m=max(m, lcm(Vec(p))))); m} \\ Andrew Howroyd, Jul 03 2018

a(n)=max{ A000793(n-2), A051704(n-1), A051704(n) }, a(0)=a(1)=1.

A129647 Largest order of a permutation of n elements with exactly 2 cycles. Also the largest LCM of a 2-partition of n.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 12, 15, 20, 21, 30, 35, 42, 45, 56, 63, 72, 77, 90, 99, 110, 117, 132, 143, 156, 165, 182, 195, 210, 221, 240, 255, 272, 285, 306, 323, 342, 357, 380, 399, 420, 437, 462, 483, 506, 525, 552, 575, 600, 621, 650, 675, 702, 725, 756, 783, 812, 837

Offset: 1

Nickolas Reynolds (nickels(AT)gmail.com), Apr 25 2007

a(n) is asymptotic to (n^2)/4.

a(n) = A116921(n)*A116922(n). - Mamuka Jibladze, Aug 22 2019

			a(26) = 165 because 26 = 11+15 and lcm(11,15) = 165 is maximal.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to lcm's
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A000793, A129651.
Cf. A116921, A116922.
Maximal LCM of k positive integers with sum n for k = 2..7: this sequence (k=2), A129648 (k=3), A129649 (k=4), A129650 (k=5), A355367 (k=6), A355403 (k=7).

a:= n-> `if`(n<2, 0, max(seq(ilcm(i, n-i), i=1..n/2))):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 16 2013

Join[{0}, Rest[With[{n = 60}, Max[LCM @@@ IntegerPartitions[#, {2}]] & /@ Range[1, n]]]] (* Modified by Philip Turecek, Mar 25 2023 *)
a[n_] := If[n<2, 0, Max[Table[LCM[i, n-i], {i, 1, n/2}]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

G.f.: t^2*(1 + 2*t^3 - 5*t^4 + 8*t^5 - 4*t^6)/((1-t)^2*(1-t^4)). - Mamuka Jibladze, Aug 22 2019

cp's OEIS Frontend

A009490 Number of distinct orders of permutations of n objects; number of nonisomorphic cyclic subgroups of symmetric group S_n.

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A222029 Triangle of number of functions in a size n set for which the sequence of composition powers ends in a length k cycle.

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A060014 Sum of orders of all permutations of n letters.

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A225632 Irregular table read by rows: n-th row gives distinct values of successively iterated Landau-like functions for n, starting with the initial value 1.

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A319055 Maximum product of an integer partition of n with relatively prime parts.

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A057740 Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).

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A256067 Irregular table T(n,k): the number of partitions of n where the least common multiple of all parts equals k.

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