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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Alexander Gruber, Feb 07 2013
EXTENSIONS
More terms from Alois P. Heinz, Feb 11 2013
STATUS
approved

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Last modified April 21 23:01 EDT 2024. Contains 371886 sequences. (Running on oeis4.)