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

2,3

Links

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000005, A003415.

Sequence in context: A008296 A351397 A140185 * A106790 A078897 A322034

Adjacent sequences: A229338 A229339 A229340 * A229342 A229343 A229344

Keyword

nonn

Author

Luca Brigada Villa, Sep 24 2013

Status

approved