A068327 Arithmetic derivative of n^n.

0, 0, 4, 27, 1024, 3125, 233280, 823543, 201326592, 2324522934, 70000000000, 285311670611, 142657607172096, 302875106592253, 100008061430022144, 3503151123046875000, 590295810358705651712, 827240261886336764177, 826274569581227289083904, 1978419655660313589123979

Offset: 0

Author

Reinhard Zumkeller, Feb 27 2002

Keywords

• nonn
• nice

Comments

p prime: a(p) = A003415(p^p) = p^p.

A003415(A051674(n)) = A051674(n).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..380 (first 100 terms from T. D. Noe)

Crossrefs

Cf. A000312, A003415.

Programs

Maple

a:= n-> n^(n+1)*add(i[2]/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 09 2015
# alternative
A068327 := proc(n)
        A003415(n^n) ;
end proc:
seq( A068327(n),n=0..10) ; # R. J. Mathar, Oct 19 2021

Mathematica

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 *)

Python

from sympy import factorint
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

Formula

a(n) = A003415(A000312(n)).

a(n) = n^n * A003415(n) = A000312(n) * A003415(n). - Alois P. Heinz, Jun 09 2015