A068237 Numerators of arithmetic derivative of 1/n: -A003415(n)/n^2.

0, -1, -1, -1, -1, -5, -1, -3, -2, -7, -1, -1, -1, -9, -8, -1, -1, -7, -1, -3, -10, -13, -1, -11, -2, -15, -1, -2, -1, -31, -1, -5, -14, -19, -12, -5, -1, -21, -16, -17, -1, -41, -1, -3, -13, -25, -1, -7, -2, -9, -20, -7, -1, -1, -16, -23, -22, -31, -1, -23

Offset: 1

Author

Reinhard Zumkeller, Feb 23 2002

Keywords

• sign
• frac
• look

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003415, A068238 (denominators).

Programs

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= n-> numer(-d(n)/n^2):
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 07 2015

Mathematica

d[n_] := If[n < 2, 0, n Sum[f[[2]]/f[[1]], {f, FactorInteger[n]}]];
a[n_] := Numerator[-d[n]/n^2];
Array[a, 80] (* Jean-François Alcover, Mar 12 2019 *)

Python

from fractions import Fraction
from sympy import factorint
def A068237(n): return -Fraction(sum((Fraction(e,p) for p,e in factorint(n).items())),n).numerator # Chai Wah Wu, Nov 03 2022