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Divisor function applied thrice to n!.
1

%I #13 Aug 12 2024 04:24:41

%S 1,2,2,3,2,4,6,6,6,5,8,8,8,8,8,9,8,10,8,10,10,12,10,10,10,15,16,16,18,

%T 16,8,16,10,12,16,18,14,3,14,16,24,16,16,20,24,24,28,24,24,16,24,24,

%U 32,32,28,32,18,20,24,28,36,32,36,24,21,24,20,40,40,30,30,36,40,42,24,32

%N Divisor function applied thrice to n!.

%H Amiram Eldar, <a href="/A053023/b053023.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = d(d(d(n!))) = A036450(A000142(n)) = A036450(n!).

%e A036450(x) = 8 if x = 11!, 12!, 13!, 14!, 15!. The reversed sequences of these d-iterates are {2, 3, 4, 8, 24, 540, 39916800}, {2, 3, 4, 8, 24, 792, 479001600}, {2, 3, 4, 8, 30, 1584, 6227020800}, {2, 3, 4, 8, 30, 2592, 87178291200}, {2, 3, 4, 8, 42, 4032, 1307674368000}.

%t a[n_] := Nest[DivisorSigma[0, #]&, n!, 3]; Array[a, 100] (* _Amiram Eldar_, Aug 12 2024 *)

%o (PARI) a(n) = numdiv(numdiv(numdiv(n!))); \\ _Amiram Eldar_, Aug 12 2024

%Y Cf. A000005, A000142, A010553, A036450, A053021.

%K nonn

%O 1,2

%A _Labos Elemer_, Feb 24 2000