A099310 Arithmetic derivative of Euler's totient function phi(n).

0, 0, 1, 1, 4, 1, 5, 4, 5, 4, 7, 4, 16, 5, 12, 12, 32, 5, 21, 12, 16, 7, 13, 12, 24, 16, 21, 16, 32, 12, 31, 32, 24, 32, 44, 16, 60, 21, 44, 32, 68, 16, 41, 24, 44, 13, 25, 32, 41, 24, 80, 44, 56, 21, 68, 44, 60, 32, 31, 32, 92, 31, 60, 80, 112, 24, 61, 80, 48, 44, 59, 44, 156, 60

Offset: 1

Author

T. D. Noe, Oct 12 2004

Keywords

• nonn

References

• See A003415

Links

• Michel Marcus, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A003415 (arithmetic derivative of n).

Programs

GAP

A099310:= Concatenation([0,0],List(List(List([3..10^3], n->Phi(n)),Factors),i->Product(i)*Sum(i,j->1/j))); # Muniru A Asiru, Sep 27 2017

Mathematica

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[dn[EulerPhi[n]], {n, 100}]

PARI

ad(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
a(n) = ad(eulerphi(n)); \\ Michel Marcus, Sep 29 2017

Formula

a(n) = A003415(A000010(n))