A211391 The number of divisors d of n! such that d < A000793(n) (Landau's function g(n)) and the symmetric group S_n contains no elements of order d.

0, 0, 0, 0, 0, 0, 2, 2, 2, 6, 4, 15, 15, 24, 29, 33, 63, 55, 126, 117, 110, 103, 225, 212, 288, 282, 319, 428, 504, 774, 859, 943, 924, 1336, 1307, 1681, 1869, 2097, 2067, 2866, 3342, 3487, 5612, 5567, 5513, 5549, 9287, 9220, 11594, 11524, 11481, 11403, 18690

Offset

1,7

Comments

This sequence gives the number of divisors d of |S_n| such that d < Lambda(n) (where Lambda(n) = the largest order of an element in S_n) for which S_n contains no element of order d. These divisors constitute a set of 'missing' element orders of S_n.

For computational purposes, the smallest divisor d0(n) of n! = |S_n| for which S_n has no element of order d0(n) is the smallest divisor of n! which is not the least common multiple of an integer partition of n. Thus d0(n) is given by the smallest prime power >= n+1 that is not prime (with the exception of n = 3 and 4, for which d0(n) = 6).

Links

Table of n, a(n) for n=1..53.

Example

For n = 7, we refer to the following table:
Symmetric Group on 7 letters.
  # of elements of order  1 ->    1
  # of elements of order  2 ->  231
  # of elements of order  3 ->  350
  # of elements of order  4 ->  840
  # of elements of order  5 ->  504
  # of elements of order  6 -> 1470
  # of elements of order  7 ->  720
  # of elements of order  8 ->    0
  # of elements of order  9 ->    0
  # of elements of order 10 ->  504
  # of elements of order 12 ->  420
  (All other divisors of 7! -> 0.)
So there are two missing element orders in S_7, whence a(7) = 2.

Prog

(Magma)
for n in [1..25] do
D := Set(Divisors(Factorial(n)));
O := { LCM(s) : s in Partitions(n) };
L := Max(O);
N := D diff O;
#{ n : n in N | n lt L };
end for;

Crossrefs

d0(n) is equal to A167184(n) for n >= 5.

Cf. A000793 (Landau's function g(n)), A057731, A211392.

Sequence in context: A300413 A246707 A324339 * A309078 A241543 A210740

Adjacent sequences: A211388 A211389 A211390 * A211392 A211393 A211394

Keyword

nonn

Author

Alexander Gruber, Feb 07 2013

Extensions

More terms from Alois P. Heinz, Feb 11 2013

Status

approved