A074859 -id:A074859 - cp's OEIS frontend

A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

1, 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, 30, 60, 60, 84, 105, 140, 210, 210, 420, 420, 420, 420, 840, 840, 1260, 1260, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 13860, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060, 60060, 120120

Offset: 0

N. J. A. Sloane

Also the largest orbit size (cycle length) for the permutation A057511 acting on Catalan objects (e.g., planar rooted trees, parenthesizations). - Antti Karttunen, Sep 07 2000
Grantham mentions that he computed a(n) for n <= 500000.
An easy lower bound is a(n) >= A002110(max{ m | A007504(m) <= n}), with strict inequality if n is not in A007504 (sum of the first m primes). Indeed, if A007504(m) <= n, the partition of n into the first m primes and maybe one additional term will have an LCM greater than or equal to primorial(m). If n > A007504(m) then a(n) >= (3/2)*A002110(m) by replacing the initial 2 by 3. But even for n = A007504(m), one has a(n) > A002110(m) for m > 8, since replacing 2+23 in 2+3+5+7+11+13+17+19+23 by 16+9, one has an LCM of 8*3*primorial(8) > primorial(9) because 24 > 23. - M. F. Hasler, Mar 29 2015
Maximum degree of the splitting field of a polynomial of degree n over a finite field, since over a finite field the degree of the splitting field is the least common multiple of the degrees of the irreducible polynomial factors of the polynomial. - Charles R Greathouse IV, Apr 27 2015
Maximum order of the elements in the symmetric group S_n. - Jianing Song, Dec 12 2021

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 6*x^6 + 12*x^7 + 15*x^8 + ...
From _Joerg Arndt_, Feb 15 2013: (Start)
The 15 partitions of 7 are the following:
[ #]  [ partition ]   lcm( parts )
[ 1]  [ 1 1 1 1 1 1 1 ]   1
[ 2]  [ 1 1 1 1 1 2 ]   2
[ 3]  [ 1 1 1 1 3 ]   3
[ 4]  [ 1 1 1 2 2 ]   2
[ 5]  [ 1 1 1 4 ]   4
[ 6]  [ 1 1 2 3 ]   6
[ 7]  [ 1 1 5 ]   5
[ 8]  [ 1 2 2 2 ]   2
[ 9]  [ 1 2 4 ]   4
[10]  [ 1 3 3 ]   3
[11]  [ 1 6 ]   6
[12]  [ 2 2 3 ]   6
[13]  [ 2 5 ]  10
[14]  [ 3 4 ]  12  (max)
[15]  [ 7 ]   7
The maximum (LCM) value attained is 12, so a(7) = 12.
(End)

J. Haack, "The Mathematics of Steve Reich's Clapping Music," in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Edmund Georg Hermann Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, p. 223.
J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdős I.
S. M. Shah, An inequality for the arithmetical function g(x), J. Indian Math. Soc., 3 (1939), 316-318. [See below for a scan of the first page.]
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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 0..814 from David Wasserman)
Giedrius Alkauskas, Colouring monohedral tilings: defects and grain boundaries, 2024.
Joerg Arndt, Table of n, a(n) for n = 0..65536 (xz compressed).
Jan Brandts and A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, arXiv:0803.2160 [math.NT], 2008.
Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
Marc Deléglise and Jean-Louis Nicolas, The Landau function and the Riemann hypothesis, Univ. Lyon (France, 2019).
John D. Dixon and Daniel Panario, The Degree of the Splitting Field of a Random Polynomial over a Finite Field, The Electronic Journal of Combinatorics 11:1 (2004).
FindStat - Combinatorial Statistic Finder, The order of a permutation
Jon Grantham, The largest prime dividing the maximal order of an element of S_n, Math. Comput. 64, No. 209, 407-410 (1995).
Joel K. Haack, The Mathematics of Steve Reich's Clapping Music,in Bridges: Mathematical Connections in Art, Music and Science: Conference Proceedings, 1998, Reza Sarhangi (ed.), 87-92.
Jan Elmar Krauskopf, Andreas Rauh, and Andreas Hein, Discrete simulation of maypole braiding machines to create collision-free braiding programmes, Heliyon (2025), Art. No. e42917.
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, (French) [Explicit upper bound on the maximum order of an element of the symmetric group] Ann. Fac. Sci. Toulouse Math. (5) 6 (1984), no. 3-4, 269--281 (1985). MR0799599 (87a:11093).
Jean-Pierre Massias, Jean-Louis Nicolas, and Guy Robin, Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53 (1989), no. 188, 665--678. MR0979940 (90e:11139).
W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
Jean-Louis Nicolas, Sur l'ordre maximum d'un élément dans le groupe Sn des permutations, Acta Arith. 14, 315-332 (1968).
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n des permutations et 'highly composite numbers', Bull. Soc. Math. France 97 (1969), 129-191.
Jean-Louis Nicolas, Calcul de l'ordre maximum d'un élément du groupe symétrique S_n Revue française d'informatique et de recherche opérationnelle, série rouge 3.2 (1969): 43-50.
Jean-Louis Nicolas, Ordre maximal d'un e'le'ment du'un groupe de permutations, C. R. Acad. Sci. Paris, A, 270 (1970), 1-4. [Annotated scanned copy]
Roger D. Nussbaum, Lunel Verduyn, and M. Sjoerd, Asymptotic estimates for the periods of periodic points of non-expansive maps, Ergodic Theory Dynam. Systems 23 (2003), no. 4, pp. 1199-1226. MR1997973 (2004m:37033).
S. M. Shah, An inequality for the arithmetical function g(x) (scan of first page).
A. Wechsler, Re: Question (A000793(A007504(n)) =? A002110(n)), SeqFan mailing list, Mar 29 2015.
Eric Weisstein's World of Mathematics, Landau's Function
Herbert S. Wilf, The asymptotics of e^P(z) and the number of elements of each order in S_n, Bull. Amer. Math. Soc., 15.2 (1986), 225-232.
Index entries for sequences related to lcm's
Index entries for "core" sequences

Cf. A000792, A009490, A034891, A057731, A074859, A128305, A129759, A225655, A225648-A225650, A225651, A225636, A225558.

Row 1 of A225630.

a000793 = maximum . map (foldl lcm 1) . partitions where
   partitions n = ps 1 n where
      ps x 0 = [[]]
      ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Mar 29 2015

A000793 := proc(n)
    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]))
        end if:
    end do:
    l ;
end proc:
seq(A000793(n),n=0..30) ; # James Sellers, Dec 07 2000
seq( max( op( map( x->ilcm(op(x)), combinat[partition](n)))), n=0..30); # David Radcliffe, Feb 28 2006
# third Maple program:
b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, max(b(n, i-1),
           seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
a:=n->b(n, `if`(n<8, 3, numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 16 2013

f[n_] := Max@ Apply[LCM, IntegerPartitions@ n, 1]; Array[f, 47] (* Robert G. Wilson v, Oct 23 2011 *)
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}]]]]; a[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 07 2014, after Alois P. Heinz *)

{a(n) = my(m, t, j, u); if( n<2, n>=0, m = ceil(n / exp(1)); t = ceil( (n/m)^m ); j=1; for( i=2, t, u = factor(i); u = sum( k=1, matsize(u)[1], u[k,1]^u[k,2]); if( u<=n, j=i)); j)}; /* Michael Somos, Oct 20 2004 */

c=0;A793=apply(t->eval(concat(Vec(t)[#Str(c++) .. -1])),select(t->#t,readstr("/tmp/b000793.txt")));A000793(n)=A793[n+1] \\ Assumes the b-file in the /tmp (or C:\tmp) folder. - M. F. Hasler, Mar 29 2015

A008475(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]^f[i,2]);
a(n)=
{
  if(n<2, return(1));
  forstep(i=ceil(exp(1.05315*sqrt(log(n)*n))), 2, -1,
    if(A008475(i)<=n, return(i))
  );
  1;
} \\ Charles R Greathouse IV, Apr 28 2015

{ \\ translated from code given by Tomas Rokicki
  my( N = 100 );
  my( V = vector(N,j,1) );
   forprime (i=2, N,  \\ primes i
      forstep (j=N, i,  -1,
         my( hi = V[j] );
         my( pp = i );  \\ powers of prime i
         while ( pp<=j,  \\ V[] is 1-based
             hi = max(if(j==pp, pp, V[j-pp]*pp), hi);
             pp *= i;
         );
         V[j] = hi;
      );
   );
   print( V );  \\ all values
\\   print( V[N] );  \\ just a(N)
\\  print("0 1");  for (n=1, N, print(n, " ", V[n]) );  \\ b-file
} \\ Joerg Arndt, Nov 14 2016

{a(n) = my(m=1); if( n<0, 0, forpart(v=n, m = max(m, lcm(Vec(v)))); m)}; /* Michael Somos, Sep 04 2017 */

from sympy import primerange
def aupton(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    for i in primerange(2, N+1):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max((pp if j==pp else V[j-pp]*pp), hi)
                pp *= i
            V[j] = hi
    return V
print(aupton(47)) # Michael S. Branicky, Oct 09 2022 after Joerg Arndt

from sympy import primerange,sqrt,log,Rational
def f(N): # compute terms a(0)..a(N)
    V = [1 for j in range(N+1)]
    if N < 4:
        C = 2
    else:
        C = Rational(166,125)
    for i in primerange(C*sqrt(N*log(N))):
        for j in range(N, i-1, -1):
            hi = V[j]
            pp = i
            while pp <= j:
                hi = max(V[j-pp]*pp, hi)
                pp *= i
            V[j] = hi
    return V
# Philip Turecek, Mar 31 2023

def a(n):
  return max([lcm(l) for l in Partitions(n)])
# Philip Turecek, Mar 28 2023

;; A naive algorithm searching through all partitions of n:
(define (A000793 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n 1 lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
(define (fold_over_partitions_of m initval addpartfun colfun) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))
;; From Antti Karttunen, May 17 2013.

Landau: lim_{n->oo} (log a(n)) / sqrt(n log n) = 1.
For bounds, see the Shah and Massias references.
For n >= 2, a(n) = max_{k} A008475(k) <= n. - Joerg Arndt, Nov 13 2016

More terms from David W. Wilson

Removed erroneous comment about a(16) which probably originated from misreading a(15)=105 as a(16) because of offset=0: a(16) = 4*5*7 = 140 is correct as it stands. - M. F. Hasler, Feb 02 2009

A057731 Irregular triangle read by rows: T(n,k) = number of elements of order k in symmetric group S_n, for n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 20, 30, 24, 20, 1, 75, 80, 180, 144, 240, 1, 231, 350, 840, 504, 1470, 720, 0, 0, 504, 0, 420, 1, 763, 1232, 5460, 1344, 10640, 5760, 5040, 0, 4032, 0, 3360, 0, 0, 2688, 1, 2619, 5768, 30996, 3024, 83160, 25920, 45360, 40320, 27216, 0, 30240, 0, 25920, 24192, 0, 0, 0, 0, 18144

Offset: 1

Roger Cuculière, Oct 29 2000

Every row for n >= 7 contains zeros. Landau's function quickly becomes > 2*n, and there is always a prime between n and 2*n. T(n,p) = 0 for such a prime p. - Franklin T. Adams-Watters, Oct 25 2011

			Triangle begins:
  1;
  1,   1;
  1,   3,   2;
  1,   9,   8,   6;
  1,  25,  20,  30,  24,   20;
  1,  75,  80, 180, 144,  240;
  1, 231, 350, 840, 504, 1470, 720, 0, 0, 504, 0, 420;
  ...

Herbert S. Wilf, "The asymptotics of e^P(z) and the number of elements of each order in S_n." Bull. Amer. Math. Soc., 15.2 (1986), 225-232.

Alois P. Heinz, Rows n = 1..30, flattened
FindStat - Combinatorial Statistic Finder, The order of a permutation.
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). - _N. J. A. Sloane_, Mar 27 2015

Cf. A000793, also A054522 (for cyclic group), A057740 (alternating group), A057741 (dihedral group).
Rows sums give A000142, last elements of rows give A074859, columns k=2, 3, 5, 7, 11 give A001189, A001471, A059593, A153760, A153761. - Alois P. Heinz, Feb 16 2013
Main diagonal gives A074351.
Cf. A222029.

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

with(group):
for n from 1 do
    f := [seq(0,i=1..n!)] ;
    mknown := 0 ;
    # loop through the permutations of n
    Sn := combinat[permute](n) ;
    for per in Sn do
        # write this permutation in cycle notation
        gen := convert(per,disjcyc) ;
        # compute the list of lengths of the cycles, then the lcm of these
        cty := [seq(nops(op(i,gen)),i=1..nops(gen))] ;
        if cty <> [] then
            lcty := lcm(op(cty)) ;
        else
            lcty := 1 ;
        end if;
        f := subsop(lcty = op(lcty,f)+1,f) ;
        mknown := max(mknown,lcty) ;
    end do:
    ff := add(el,el=f) ;
    print(seq(f[i],i=1..mknown)) ;
end do: # R. J. Mathar, May 26 2014
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Jul 11 2017

<Jean-François Alcover, Aug 31 2016 *)
b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j-1)!*b[n-j, LCM[g, j]]* Binomial[n-1, j-1], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n, 1]];
Array[T, 12] // Flatten (* Jean-François Alcover, May 03 2019, after Alois P. Heinz *)

T(n,k)={n!*polcoeff(sumdiv(k, i, moebius(k/i)*exp(sumdiv(i, j, x^j/j) + O(x*x^n))), n)} \\ Andrew Howroyd, Jul 02 2018

Sum_{k=1..A000793(n)} k*T(n,k) = A060014(n); A000793 = Landau's function.

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

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

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

cp's OEIS Frontend

A074103 a(n) = n!/A074859(n).

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A000793 Landau's function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n.

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A057731 Irregular triangle read by rows: T(n,k) = number of elements of order k in symmetric group S_n, for n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

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A060014 Sum of orders of all permutations of n letters.

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

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A074351 Number of elements of S_n having order n.