%I #31 Jun 13 2022 15:16:26

%S 0,0,4,216,165888,604800000,48372940800000,43156963184025600000,

%T 1392410948543163924480000000,668916177911197542484208831692800000,

%U 8199617664717905359483850194944000000000000000,2401010998878767104110478543683244630474752000000000000000

%N Arithmetic derivative of hyperfactorial(n).

%H Alois P. Heinz, <a href="/A260619/b260619.txt">Table of n, a(n) for n = 0..37</a>

%F a(n) = A003415(A002109(n)).

%F a(n) = A002109(n)*A190121(n) (conjectured).

%p h:= proc(n) option remember; `if`(n=0, 1, h(n-1)* n^n) end:

%p a:= proc(n) n^n *`if`(n=0, 0,

%p a(n-1)+h(n-1)*n*add(i[2]/i[1], i=ifactors(n)[2]))

%p end:

%p seq(a(n), n=0..15); # _Alois P. Heinz_, Sep 18 2015

%t a[n_] := If[n<2, 0, With[{h = Hyperfactorial[n]}, h Sum[{p, e} = pe; e/p, {pe, FactorInteger[h]}]]];

%t a /@ Range[0, 15] (* _Jean-François Alcover_, Nov 14 2020 *)

%o (Python 3.8+)

%o from math import prod

%o from collections import Counter

%o from sympy import factorint

%o def A260619(n):

%o s = prod(i**i for i in range(2,n+1))

%o 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 Cf. A002109, A003415, A068327.

%K nonn

%O 0,3

%A _Matthew Campbell_, Sep 17 2015

%E More terms from _Alois P. Heinz_, Sep 18 2015