A000793 -id:A000793 - cp's OEIS frontend

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.

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

A074859 Number of elements of S_n having the maximum possible order g(n), where g(n) is Landau's function (A000793).

Original entry on oeis.org

1, 1, 1, 2, 6, 20, 240, 420, 2688, 18144, 120960, 2661120, 7983360, 103783680, 1037836800, 12454041600, 149448499200, 1693749657600, 60974987673600, 289631191449600, 5792623828992000, 121645100408832000, 3568256278659072000, 30776210403434496000, 738629049682427904000, 12310484161373798400000

Offset: 0

Christopher J. Smyth, Sep 11 2002

J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdős I.

Alois P. Heinz, Table of n, a(n) for n = 0..175
J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
J.-L. Nicolas, Sur l'ordre maximum d'un élément dans le groupe S_n des permutations, Acta Arith., 14 (1968), 315-332.
J.-L. Nicolas, Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers', Bull. Math. Soc. France 97 (1969) 129-191.

Cf. A000793 (Landau's function g(n)).
Cf. A074064, A074103, A074115, A162682.
Last row element of A057731. - Alois P. Heinz, Feb 14 2013

g[n_] := Max[ Apply[ LCM, IntegerPartitions[n], 1]]; f[x_, n_] := Total[ (MoebiusMu[g[n]/#]*Exp[ Total[ (x^#/# & ) /@ Divisors[#]]] & ) /@ Divisors[g[n]]]; a[n_] := n!*Coefficient[ Series[f[x, n], {x, 0, n}], x^n]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 03 2011, after Vladeta Jovovic *)

a(n) = n!*coefficient of x^n in expansion of Sum_{i divides A000793(n)} mu(A000793(n)/i)*exp(Sum_{j divides i} x^j/j). - Vladeta Jovovic, Sep 29 2002

Corrected and extended by Vladeta Jovovic, Sep 20 2002

A002809 Increasing values of A000793 (largest order of permutation of n elements).

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 15, 20, 30, 60, 84, 105, 140, 210, 420, 840, 1260, 1540, 2310, 2520, 4620, 5460, 9240, 13860, 16380, 27720, 30030, 32760, 60060, 120120, 180180, 360360, 471240, 510510, 556920, 1021020, 1141140, 2042040, 3063060, 3423420, 6126120, 6846840

Offset: 1

N. J. A. Sloane

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).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 3318 terms from Alois P. Heinz)
Marc Deléglise and Jean-Louis Nicolas, Maximal product of primes whose sum is bounded, arXiv preprint arXiv:1207.0603 [math.NT], 2012. - From _N. J. A. Sloane_, Dec 17 2012
J.-L. Nicolas, Sur l'ordre maximum d'un élément dans le groupe S_n des permutations, Acta Arith., 14 (1968), 315-332.
J.-L. Nicolas, Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers', Bull. Math. Soc. France 97 (1969) 129-191.

Indices are A006644.

Cf. A000793.

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}] // Union (* Jean-François Alcover, Mar 07 2014, after Alois P. Heinz *)

Description improved Apr 15 1997. More terms from David W. Wilson.

A225558 a(n) = A003418(n)/A000793(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 10, 35, 56, 126, 84, 924, 462, 6006, 4290, 3432, 5148, 58344, 58344, 554268, 554268, 554268, 554268, 6374082, 6374082, 21246940, 21246940, 52151580, 34767720, 924241890, 504131940, 15628090140, 26447537160, 26447537160, 15628090140

Offset: 0

Antti Karttunen, May 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000793, A003418.

g:= proc(n) g(n):= `if`(n=0, 1, ilcm(n, g(n-1))) end:
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->g(n)/b(n, `if`(n<8, 3,
    numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
seq(a(n), n=0..40); # Alois P. Heinz, May 22 2013

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[0]=1; a[n_] := LCM @@ Range[n] / b[n, If[n<8, 3, PrimePi[ Ceiling[ 1.328*Sqrt[n*Log[n] // Floor]]]]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 27 2016, after Alois P. Heinz *)

a(n) = A003418(n)/A000793(n).

A074115(n)/a(n) = A025527(n).

A225636 a(n) = A225627(n)/A000793(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 5, 7, 8, 9, 7, 14, 11, 13, 15, 44, 39, 26, 26, 22, 33, 33, 39, 39, 143, 143, 143, 153, 78, 187, 221, 221, 209, 209, 247, 247, 323, 323, 418, 418, 391, 646, 437, 437, 646, 969, 969, 969, 969, 782, 874, 874, 1292, 667, 713, 713, 782, 5681, 3496

Offset: 0

Antti Karttunen, May 13 2013

nonn

a(n) divides A225558(n) for all n.

Cf. A225637, A225652, A225558.

(define (A225636 n) (/ (A225627 n) (A000793 n)))

A225648 Positions of ones in A225650, numbers n such that n and A000793(n) are coprime.

Original entry on oeis.org

0, 1, 5, 7, 8, 9, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313

Offset: 1

Antti Karttunen, May 12 2013

nonn

Contains all primes from 5 onward. Are 8, 9 and 27 only composite numbers present?

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Complement: A225649. Cf. also A225650.

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_] := b[n, If[n < 8, 3, PrimePi[Ceiling[1.328*Sqrt[n* Log[n] // Floor]]]]]; Join[{0}, Position[Table[GCD[n, g[n]], {n, 1, 500} ], 1] // Flatten] (* Jean-François Alcover, Mar 03 2016, after Alois P. Heinz *)

A225651 Numbers k that divide A000793(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 14, 15, 20, 21, 24, 30, 35, 36, 39, 40, 42, 44, 52, 55, 56, 60, 65, 66, 70, 72, 76, 77, 78, 84, 85, 90, 91, 95, 99, 102, 105, 110, 114, 115, 117, 119, 120, 126, 130, 132, 133, 136, 138, 140, 143, 152, 153, 154, 155, 156, 161, 165, 170

Offset: 1

Antti Karttunen, May 16 2013

nonn

After 1, a subset of A225649.

Also, for all n, A225650(a(n)) = a(n) and A225655(a(n)) = A000793(a(n)).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A225648, A225649, A225650, A225653, A225655-A225657.

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:
g:=n->b(n, `if`(n<8, 3, numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while not irem(g(k), k)=0 do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, May 22 2013

Reap[For[n=1, n <= 40, n++, If[Divisible[Max[LCM @@@ IntegerPartitions[n] ], n], Sow[n]]]][[2, 1]]
(* or, for a large number of terms: *)
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_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Reap[For[k=1, k <= 1000, k++, If[Divisible[g[k], k], Sow[ k]]]][[2, 1]] (* Jean-François Alcover, Feb 28 2016, after Alois P. Heinz *)

A225649 Positions of non-ones in A225650, numbers n such that n and A000793(n) have at least one common divisor > 1.

Original entry on oeis.org

2, 3, 4, 6, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92

Offset: 1

Antti Karttunen, May 13 2013

nonn

Which composites are missing apart from 8, 9 and 27? See comment at A225648.

Cf. A225648 (complement), A225651 (from n>1 onward is a subset).

Cf. A225650, A225640.

A074115 a(n) = n!/A000793(n).

Original entry on oeis.org

1, 1, 1, 2, 6, 20, 120, 420, 2688, 18144, 120960, 1330560, 7983360, 103783680, 1037836800, 12454041600, 149448499200, 1693749657600, 30487493836800, 289631191449600, 5792623828992000, 121645100408832000, 2676192208994304000, 30776210403434496000

Offset: 0

Vladeta Jovovic, Sep 16 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..458

Cf. A074859.

a(0)=1 prepended by Alois P. Heinz, Jul 12 2017

A074881 Triangle T(n,k) giving number of labeled cyclic subgroups of order k in symmetric group S_n, 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, 1, 1, 9, 4, 3, 1, 25, 10, 15, 6, 10, 1, 75, 40, 90, 36, 120, 1, 231, 175, 420, 126, 735, 120, 126, 105, 1, 763, 616, 2730, 336, 5320, 960, 1260, 1008, 840, 336, 1, 2619, 2884, 15498, 756, 41580, 4320, 11340, 6720, 6804, 7560, 4320, 3024, 2268

Offset: 1

Vladeta Jovovic, Sep 30 2002

A057731 contains zeros. This sequence contains only positive values of A057731(n,k)/A000010(k). - Alois P. Heinz, Feb 16 2013

			Triangle begins:
  1;
  1,   1;
  1,   3,   1;
  1,   9,   4,   3;
  1,  25,  10,  15,   6,  10;
  1,  75,  40,  90,  36, 120;
  1, 231, 175, 420, 126, 735, 120, 126, 105;
  ...

Alois P. Heinz, Rows n = 1..42, flattened

Row sums give A051625.

Cf. A000010, A181949.

nmax = 10;
T[n_, k_] := n! SeriesCoefficient[O[x]^(n+1) + Sum[MoebiusMu[k/i]*Exp[ Sum[x^j/j, {j, Divisors[i]}]], {i, Divisors[k]}], {x, 0, n}]/ EulerPhi[k];
Table[DeleteCases[Table[T[n, k], {k, 1, 2 nmax}], 0], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Sep 16 2019, after Andrew Howroyd *)

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

T(n,k) = A057731(n,k)/A000010(k).

cp's OEIS Frontend

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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A074859 Number of elements of S_n having the maximum possible order g(n), where g(n) is Landau's function (A000793).

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A002809 Increasing values of A000793 (largest order of permutation of n elements).

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A225558 a(n) = A003418(n)/A000793(n).

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A225636 a(n) = A225627(n)/A000793(n).

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A225648 Positions of ones in A225650, numbers n such that n and A000793(n) are coprime.

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A225651 Numbers k that divide A000793(k).

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A225649 Positions of non-ones in A225650, numbers n such that n and A000793(n) have at least one common divisor > 1.

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