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A260619 Arithmetic derivative of hyperfactorial(n).

%I A260619 #31 Jun 13 2022 15:16:26
%S A260619 0,0,4,216,165888,604800000,48372940800000,43156963184025600000,
%T A260619 1392410948543163924480000000,668916177911197542484208831692800000,
%U A260619 8199617664717905359483850194944000000000000000,2401010998878767104110478543683244630474752000000000000000
%N A260619 Arithmetic derivative of hyperfactorial(n).
%H A260619 Alois P. Heinz, <a href="/A260619/b260619.txt">Table of n, a(n) for n = 0..37</a>
%F A260619 a(n) = A003415(A002109(n)).
%F A260619 a(n) = A002109(n)*A190121(n) (conjectured).
%p A260619 h:= proc(n) option remember; `if`(n=0, 1, h(n-1)* n^n) end:
%p A260619 a:= proc(n) n^n *`if`(n=0, 0,
%p A260619       a(n-1)+h(n-1)*n*add(i[2]/i[1], i=ifactors(n)[2]))
%p A260619     end:
%p A260619 seq(a(n), n=0..15);  # _Alois P. Heinz_, Sep 18 2015
%t A260619 a[n_] := If[n<2, 0, With[{h = Hyperfactorial[n]}, h Sum[{p, e} = pe; e/p, {pe, FactorInteger[h]}]]];
%t A260619 a /@ Range[0, 15] (* _Jean-François Alcover_, Nov 14 2020 *)
%o A260619 (Python 3.8+)
%o A260619 from math import prod
%o A260619 from collections import Counter
%o A260619 from sympy import factorint
%o A260619 def A260619(n):
%o A260619     s = prod(i**i for i in range(2,n+1))
%o A260619     return sum(s*e//p for p,e in sum(((lambda x: Counter({k:x[k]*m 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 A260619 Cf. A002109, A003415, A068327.
%K A260619 nonn
%O A260619 0,3
%A A260619 _Matthew Campbell_, Sep 17 2015
%E A260619 More terms from _Alois P. Heinz_, Sep 18 2015