A068329 Arithmetic derivative of Fibonacci numbers > 0.

0, 0, 1, 1, 1, 12, 1, 10, 19, 16, 1, 384, 1, 42, 437, 491, 1, 4164, 150, 4388, 6341, 288, 1, 155472, 30035, 754, 115271, 142474, 1, 1530588, 2974, 1084624, 1802069, 5168, 2555363, 46594656, 503939, 1406531

Offset: 1

Author

Reinhard Zumkeller, Feb 27 2002

Keywords

• nonn

Links

• Iain Fox, Table of n, a(n) for n = 1..400 (first 200 terms from T. D. Noe)

Crossrefs

Cf. A000045, A003415, A001605 (where a(n) = 1).

Programs

GAP

a:=Concatenation([0,0],List(List([3..40],n->Factors(Fibonacci(n))),i->Product(i)*Sum(i,j->1/j))); # Muniru A Asiru, Oct 31 2018

Magma

Ad:=func; [n le 2 select 0 else Ad(Fibonacci(n)): n in [1..40]]; // Bruno Berselli, Oct 22 2013

Mathematica

ad[1] = 0; ad[n_] := Module[{f = FactorInteger[n]}, Total[n*f[[All, 2]]/ f[[All, 1]]]]; a[n_] := ad[Fibonacci[n]]; Array[a, 40] (* Jean-François Alcover, Feb 22 2018 *)

PARI

a(n) = my(f = factor(n=fibonacci(n))~); sum(i=1, #f, n/f[1, i]*f[2, i]) \\ Iain Fox, Oct 29 2018

Python

from sympy import fibonacci, factorint
def A068329(n):
    f = fibonacci(n)
    return sum((f*e//p for p,e in factorint(f).items())) if n > 2 else 0 # Chai Wah Wu, Jun 12 2022

Formula

a(n) = A003415(A000045(n)).