A229341 a(n) = tau(n'), the number of divisors of the arithmetic derivative of n.

1, 1, 3, 1, 2, 1, 6, 4, 2, 1, 5, 1, 3, 4, 6, 1, 4, 1, 8, 4, 2, 1, 6, 4, 4, 4, 6, 1, 2, 1, 10, 4, 2, 6, 12, 1, 4, 5, 6, 1, 2, 1, 10, 4, 3, 1, 10, 4, 6, 6, 8, 1, 5, 5, 6, 4, 2, 1, 6, 1, 4, 4, 14, 6, 2, 1, 12, 4, 2, 1, 12, 1, 4, 4, 10, 6, 2, 1, 10, 12, 2, 1, 6, 4, 6

Offset: 2

Author

Luca Brigada Villa, Sep 24 2013

Keywords

• nonn

Examples

			For n=4, tau(n')=tau(4)=3.
For n=5, tau(n')=tau(1)=1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 2..65537

Crossrefs

Cf. A000005, A003415.

Programs

GAP

List(List(List([2..10^2],Factors),i->Product(i)*Sum(i,j->1/j)),Tau); # Muniru A Asiru, Mar 05 2018

Mathematica

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose@ FactorInteger@ n}, If[PrimeQ@n, 1, Total[n*f[[2]]/f[[1]]]]]; (* see A003415 *); f[n_] := DivisorSigma[0, dn@ n]; Array[f, 85, 2] (* Robert G. Wilson v, Mar 12 2018 *)

PARI

rd(n) = {local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1];));}
a(n) = numdiv(rd(n)); \\ Michel Marcus, Sep 24 2013

Formula

a(n) = A000005(A003415(n)).