A056044 - OEIS

%I #14 May 26 2024 08:37:20

%S 1,1,1,1,1,2,2,1,3,1,1,2,2,2,6,3,3,2,2,2,2,2,2,2,10,10,30,2,2,12,12,3,

%T 3,6,30,10,10,10,30,6,6,2,2,2,30,60,60,30,210,42,42,42,42,1,1,2,2,4,4,

%U 4,4,4,84,21,21,14,14,14,42,6,6,2,2,2,10,10,70,140,140,14,126,3,3,6,30

%N Let k be the largest number such that k^2 divides n! and let m be the largest number such that m!^2 divides n!; a(n) = k/m!.

%H Michael De Vlieger, <a href="/A056044/b056044.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000188(n!)/A056038(n).

%e For n = 11, 11! = 6! * 6! * 77, so A000188(11!) = A056038(11) = 6! and a(11) = 6!/6! = 1.

%t f[p_, e_] := p^Floor[e/2]; b[1] = 1; b[n_] := Times @@ f @@@ FactorInteger[n!];

%t c[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; (k-1)!];

%t a[n_] := b[n] / c[n]; Array[a, 100] (* _Amiram Eldar_, May 24 2024 *)

%Y Cf. A000188, A055772, A001405, A001057, A056038.

%K nonn

%O 1,6

%A _Labos Elemer_, Jul 25 2000

%E Name corrected by _Amiram Eldar_, May 24 2024