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A068327 Arithmetic derivative of n^n.

%I A068327 #42 Aug 19 2024 09:18:18
%S A068327 0,0,4,27,1024,3125,233280,823543,201326592,2324522934,70000000000,
%T A068327 285311670611,142657607172096,302875106592253,100008061430022144,
%U A068327 3503151123046875000,590295810358705651712,827240261886336764177,826274569581227289083904,1978419655660313589123979
%N A068327 Arithmetic derivative of n^n.
%C A068327 p prime: a(p) = A003415(p^p) = p^p.
%C A068327 A003415(A051674(n)) = A051674(n).
%H A068327 Alois P. Heinz, <a href="/A068327/b068327.txt">Table of n, a(n) for n = 0..380</a> (first 100 terms from T. D. Noe)
%F A068327 a(n) = A003415(A000312(n)).
%F A068327 a(n) = n^n * A003415(n) = A000312(n) * A003415(n). - _Alois P. Heinz_, Jun 09 2015
%e A068327 a(10) = A003415(10^10) = A003415(2^10 * 5^10) = 10^10 * (10/2 + 10/5) = 10^10 * (5 + 2) = 70000000000 by formula in A003415.
%p A068327 a:= n-> n^(n+1)*add(i[2]/i[1], i=ifactors(n)[2]):
%p A068327 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 09 2015
%p A068327 # alternative
%p A068327 A068327 := proc(n)
%p A068327         A003415(n^n) ;
%p A068327 end proc:
%p A068327 seq( A068327(n),n=0..10) ; # _R. J. Mathar_, Oct 19 2021
%t A068327 a312[n_] := Sum[ StirlingS2[n, k]*n!/(n - k)!, {k, 0, n}]; a3415[n_] := With[ {fi = FactorInteger[n]}, n*Total[ fi[[All, 2]] / fi[[All, 1]] ] ]; a3415[0] = a3415[1] = 0; a[n_] := a3415[ a312[n] ]; Table[ a[n], {n, 1, 16}] (* _Jean-François Alcover_, Mar 27 2013 *)
%o A068327 (Python)
%o A068327 from sympy import factorint
%o A068327 def A068327(n): return sum((n**(n+1)*e//p for p,e in factorint(n).items())) if n > 1 else 0 # _Chai Wah Wu_, Jun 12 2022
%Y A068327 Cf. A000312, A003415.
%K A068327 nonn,nice
%O A068327 0,3
%A A068327 _Reinhard Zumkeller_, Feb 27 2002