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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.

%I A211391 #28 Sep 08 2022 08:46:02
%S A211391 0,0,0,0,0,0,2,2,2,6,4,15,15,24,29,33,63,55,126,117,110,103,225,212,
%T A211391 288,282,319,428,504,774,859,943,924,1336,1307,1681,1869,2097,2067,
%U A211391 2866,3342,3487,5612,5567,5513,5549,9287,9220,11594,11524,11481,11403,18690
%N 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.
%C A211391 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.
%C A211391 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).
%e A211391 For n = 7, we refer to the following table:
%e A211391 Symmetric Group on 7 letters.
%e A211391   # of elements of order  1 ->    1
%e A211391   # of elements of order  2 ->  231
%e A211391   # of elements of order  3 ->  350
%e A211391   # of elements of order  4 ->  840
%e A211391   # of elements of order  5 ->  504
%e A211391   # of elements of order  6 -> 1470
%e A211391   # of elements of order  7 ->  720
%e A211391   # of elements of order  8 ->    0
%e A211391   # of elements of order  9 ->    0
%e A211391   # of elements of order 10 ->  504
%e A211391   # of elements of order 12 ->  420
%e A211391   (All other divisors of 7! -> 0.)
%e A211391 So there are two missing element orders in S_7, whence a(7) = 2.
%o A211391 (Magma)
%o A211391 for n in [1..25] do
%o A211391 D := Set(Divisors(Factorial(n)));
%o A211391 O := { LCM(s) : s in Partitions(n) };
%o A211391 L := Max(O);
%o A211391 N := D diff O;
%o A211391 #{ n : n in N | n lt L };
%o A211391 end for;
%Y A211391 d0(n) is equal to A167184(n) for n >= 5.
%Y A211391 Cf. A000793 (Landau's function g(n)), A057731, A211392.
%K A211391 nonn
%O A211391 1,7
%A A211391 _Alexander Gruber_, Feb 07 2013
%E A211391 More terms from _Alois P. Heinz_, Feb 11 2013