A068721 Arithmetic derivative of cubes: a(n) = 3*n^2*A003415(n).

0, 12, 27, 192, 75, 540, 147, 2304, 1458, 2100, 363, 6912, 507, 5292, 5400, 24576, 867, 20412, 1083, 28800, 13230, 18876, 1587, 76032, 18750, 30420, 59049, 75264, 2523, 83700, 2883, 245760, 45738, 65892, 44100

Offset: 1

Author

Reinhard Zumkeller, Feb 26 2002

Keywords

• nonn
• easy

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000578, A003415, A068720.

Programs

Magma

Ad:=func; [0] cat [Ad(n^3): n in [2..40]]; // Bruno Berselli, Oct 22 2013

Mathematica

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); ad[0] = ad[1] = 0; a[n_] := 3 * n^2 * ad[n]; Array[a, 100] (* Amiram Eldar, Apr 11 2025 *)

Python

from sympy import factorint
def A068721(n): return 3*n**2*sum((n*e//p for p,e in factorint(n).items())) if n > 1 else 0 # Chai Wah Wu, Nov 03 2022

Formula

a(n) = A003415(A000578(n)).