A073809 Number of common divisors of sigma_1(n) and sigma_3(n).

1, 2, 3, 1, 4, 6, 4, 4, 1, 6, 6, 6, 4, 8, 8, 2, 6, 2, 6, 8, 6, 9, 8, 12, 1, 8, 8, 4, 8, 12, 6, 3, 10, 8, 10, 1, 4, 12, 8, 12, 8, 12, 6, 6, 4, 12, 10, 6, 2, 2, 12, 4, 8, 16, 12, 16, 10, 12, 12, 16, 4, 12, 4, 2, 12, 15, 6, 12, 12, 15, 12, 8, 4, 8, 3, 12, 12, 16, 10, 8, 3, 12, 12, 12, 12, 12, 16

Offset: 1

Author

Labos Elemer, Aug 13 2002

Keywords

• nonn

Examples

			n=10: sigma[1,10]=18, sigma[3,10]=1134; Intersection[{1,2,3,6,9,18},{1,2,3,6,7,9,14,18,21,27,42,54,63, 81,126,162,189,378,567,1134}]={1,2,3,6,9,18}, so a(10)=6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A000203, A001158, A073802, A073808.

Programs

Mathematica

g1[x_] := Divisors[DivisorSigma[1, x]] g2[x_] := Divisors[DivisorSigma[3, x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}]
(* Second program: *)
Table[Length@ Apply[Intersection, Divisors@ Array[DivisorSigma[2 # - 1, n] &, 2]], {n, 87}] (* Michael De Vlieger, Nov 23 2017 *)

PARI

A073809(n) = numdiv(gcd(sigma(n),sigma(n,3))); \\ Antti Karttunen, Nov 23 2017

Formula

a(n) = Card[Intersection[D[A000203(n)], D[A001158(n)]]] where D[x] is the set of divisors of x.