A086134 Smallest prime factor of arithmetic derivative of n or a(n)=0 if no such prime exists.

0, 0, 0, 0, 2, 0, 5, 0, 2, 2, 7, 0, 2, 0, 3, 2, 2, 0, 3, 0, 2, 2, 13, 0, 2, 2, 3, 3, 2, 0, 31, 0, 2, 2, 19, 2, 2, 0, 3, 2, 2, 0, 41, 0, 2, 3, 5, 0, 2, 2, 3, 2, 2, 0, 3, 2, 2, 2, 31, 0, 2, 0, 3, 3, 2, 2, 61, 0, 2, 2, 59, 0, 2, 0, 3, 5, 2, 2, 71, 0, 2, 2, 43, 0, 2, 2, 3, 2, 2, 0, 3, 2, 2, 2, 7, 2, 2, 0, 7, 3, 2, 0, 7

Offset: 0

Author

Labos Elemer, Jul 23 2003

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003415.

Programs

Maple

with(numtheory):
d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= n-> (f-> `if`(f<2, 0, min(factorset(f)[])))(d(n)):
seq(a(n), n=0..105);  # Alois P. Heinz, Jun 08 2015

Mathematica

d[n_] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}];
a[n_] := Function[f, If[f<2, 0, Min[FactorInteger[f][[All, 1]]]]][d[n]]; a[0] = 0;
Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Mar 23 2017, after Alois P. Heinz *)

Python

from sympy import primefactors, factorint
def A086134(n): return 0 if n <= 1 else min(primefactors(m)) if (m:=sum((n*e//p for p,e in factorint(n).items()))) > 1 else 0 # Chai Wah Wu, Nov 04 2022