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

1,6

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)

Links

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

Formula

See program.

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

Example

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.

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 *)

Prog

(PARI) a(n)=numdiv(gcd(sigma(n), n)) \\ Charles R Greathouse IV, Mar 09 2014
(Magma) [NumberOfDivisors(GCD(SumOfDivisors(n), n)): n in [1..100]]; // Vincenzo Librandi, Oct 09 2017

Crossrefs

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

Sequence in context: A269443 A039927 A336722 * A380199 A132157 A103163

Adjacent sequences: A073799 A073800 A073801 * A073803 A073804 A073805

Keyword

nonn

Author

Labos Elemer, Aug 13 2002

Status

approved