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A260620 Arithmetic derivative of superfactorial(n).

%I A260620 #21 Jun 13 2022 13:04:08
%S A260620 0,0,1,16,912,179712,200724480,1389079756800,78810485096448000,
%T A260620 38096713995308236800000,177372596340389981454336000000,
%U A260620 8666143442523657424202209689600000000,5080543621153782266150614213475696640000000000
%N A260620 Arithmetic derivative of superfactorial(n).
%H A260620 Alois P. Heinz, <a href="/A260620/b260620.txt">Table of n, a(n) for n = 0..43</a>
%F A260620 a(n) = A003415(A000178(n)).
%p A260620 b:= proc(n) option remember; `if`(n<2, 0,
%p A260620        b(n-1)*n + n!*add(i[2]/i[1], i=ifactors(n)[2]))
%p A260620     end:
%p A260620 s:= proc(n) option remember; `if`(n=0, 1, n!*s(n-1)) end:
%p A260620 a:= proc(n) option remember; `if`(n=0, 0,
%p A260620        b(n)*s(n-1)+n!*a(n-1))
%p A260620     end:
%p A260620 seq(a(n), n=0..15);  # _Alois P. Heinz_, Sep 18 2015
%t A260620 a[n_] := If[n<2, 0, With[{g = BarnesG[n+2]}, g Sum[{p, e} = pe; e/p, {pe, FactorInteger[g]}]]];
%t A260620 a /@ Range[0, 15] (* _Jean-François Alcover_, Nov 14 2020 *)
%o A260620 (Python 3.8+)
%o A260620 from math import prod, factorial
%o A260620 from collections import Counter
%o A260620 from sympy import factorint
%o A260620 def A260620(n):
%o A260620     s = prod(factorial(i) for i in range(2,n+1))
%o A260620     return sum(s*e//p for p,e in sum(((lambda x: Counter({k:x[k]*(n-m+1) for k in x}))(factorint(m)) for m in range(2,n+1)),start=Counter({2:0})).items()) if n > 1 else 0 # _Chai Wah Wu_, Jun 12 2022
%Y A260620 Cf. A003415, A000178, A068311.
%K A260620 nonn
%O A260620 0,4
%A A260620 _Matthew Campbell_, Sep 18 2015
%E A260620 More terms from _Alois P. Heinz_, Sep 18 2015