A073802 Number of common divisors of n and sigma(n).

1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 6, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 3, 1, 1, 2, 2, 1, 4, 1, 4, 1, 2, 1, 6, 1, 2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 2, 3, 1, 6, 2, 3, 1, 2, 2, 6, 1, 1, 2, 1, 1, 4, 1, 2, 2

Offset: 1

Author

Labos Elemer, Aug 13 2002

Keywords

• nonn

Comments

From Jaroslav Krizek, Feb 18 2010: (Start)

Number of divisors d of number n such that d divides sigma(n).

a(n) = A000005(n) - A173438(n).

a(n) = A000005(n) for multiply-perfect numbers (A007691). (End)

Examples

			For n = 12: a(12) = 3; sigma(12) = 28, divisors of 12: 1, 2, 3, 4, 6, 12; d divides sigma(n) for 3 divisors d: 1, 2, 4.
n=96: d(96) = {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}, d(sigma(96)) = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, CD(n, sigma(n)) = {1, 2, 3, 4, 6, 12} so a(96) = 6.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A062068, A073803, A073804.

Programs

Magma

[NumberOfDivisors(GCD(SumOfDivisors(n),n)): n in [1..100]]; // Vincenzo Librandi, Oct 09 2017

Mathematica

g1[x_] := Divisors[x]; g2[x_] := Divisors[DivisorSigma[1, x]]; ncd[x_] := Length[Intersection[g1[x], g2[x]]]; Table[ncd[w], {w, 1, 128}]
Table[Length[Intersection[Divisors[n], Divisors[DivisorSigma[1, n]]]], {n, 100}] (* Vincenzo Librandi, Oct 09 2017 *)
a[n_] := DivisorSigma[0, GCD[n, DivisorSigma[1, n]]]; Array[a, 100] (* Amiram Eldar, Nov 21 2024 *)

PARI

a(n)=numdiv(gcd(sigma(n),n)) \\ Charles R Greathouse IV, Mar 09 2014

Formula

See program.

a(n) = A000005(A009194(n)) = tau(gcd(n,sigma(n))). [Reinhard Zumkeller, Mar 12 2010]