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A056044 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!.

%I A056044 #14 May 26 2024 08:37:20
%S A056044 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 A056044 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 A056044 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 A056044 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 A056044 Michael De Vlieger, <a href="/A056044/b056044.txt">Table of n, a(n) for n = 1..10000</a>
%F A056044 a(n) = A000188(n!)/A056038(n).
%e A056044 For n = 11, 11! = 6! * 6! * 77, so A000188(11!) = A056038(11) = 6! and a(11) = 6!/6! = 1.
%t A056044 f[p_, e_] := p^Floor[e/2]; b[1] = 1; b[n_] := Times @@ f @@@ FactorInteger[n!];
%t A056044 c[n_] := Module[{k = 1}, NestWhile[#/(++k)^2 &, n!, IntegerQ]; (k-1)!];
%t A056044 a[n_] := b[n] / c[n]; Array[a, 100] (* _Amiram Eldar_, May 24 2024 *)
%Y A056044 Cf. A000188, A055772, A001405, A001057, A056038.
%K A056044 nonn
%O A056044 1,6
%A A056044 _Labos Elemer_, Jul 25 2000
%E A056044 Name corrected by _Amiram Eldar_, May 24 2024