|
%I #36 Oct 30 2023 07:52:13
%S 0,0,4,27,1024,3125,233280,823543,201326592,2324522934,70000000000,
%T 285311670611,142657607172096,302875106592253,100008061430022144,
%U 3503151123046875000,590295810358705651712,827240261886336764177,826274569581227289083904,1978419655660313589123979
%N Arithmetic derivative of n^n.
%C p prime: a(p) = A003415(p^p)) = p^p,
%C A003415(A051674(n)) = A051674(n).
%H 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 a(n) = A003415(A000312(n)).
%F a(n) = n^n * A003415(n) = A000312(n) * A003415(n). - _Alois P. Heinz_, Jun 09 2015
%e 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 a:= n-> n^(n+1)*add(i[2]/i[1], i=ifactors(n)[2]):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 09 2015
%p # alternative
%p A068327 := proc(n)
%p A003415(n^n) ;
%p end proc:
%p seq( A068327(n),n=0..10) ; # _R. J. Mathar_, Oct 19 2021
%t 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 (Python)
%o from sympy import factorint
%o 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 Cf. A000312, A003415.
%K nonn,nice
%O 0,3
%A _Reinhard Zumkeller_, Feb 27 2002
|
