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

%I A073802 #28 Nov 21 2024 06:07:32
%S A073802 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,
%T A073802 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,
%U A073802 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
%N A073802 Number of common divisors of n and sigma(n).
%C A073802 From _Jaroslav Krizek_, Feb 18 2010: (Start)
%C A073802 Number of divisors d of number n such that d divides sigma(n).
%C A073802 a(n) = A000005(n) - A173438(n).
%C A073802 a(n) = A000005(n) for multiply-perfect numbers (A007691). (End)
%H A073802 Charles R Greathouse IV, <a href="/A073802/b073802.txt">Table of n, a(n) for n = 1..10000</a>
%F A073802 See program.
%F A073802 a(n) = A000005(A009194(n)) = tau(gcd(n,sigma(n))). [_Reinhard Zumkeller_, Mar 12 2010]
%e A073802 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.
%e A073802 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.
%t A073802 g1[x_] := Divisors[x]; g2[x_] := Divisors[DivisorSigma[1, x]]; ncd[x_] := Length[Intersection[g1[x], g2[x]]]; Table[ncd[w], {w, 1, 128}]
%t A073802 Table[Length[Intersection[Divisors[n], Divisors[DivisorSigma[1, n]]]], {n, 100}] (* _Vincenzo Librandi_, Oct 09 2017 *)
%t A073802 a[n_] := DivisorSigma[0, GCD[n, DivisorSigma[1, n]]]; Array[a, 100] (* _Amiram Eldar_, Nov 21 2024 *)
%o A073802 (PARI) a(n)=numdiv(gcd(sigma(n),n)) \\ _Charles R Greathouse IV_, Mar 09 2014
%o A073802 (Magma) [NumberOfDivisors(GCD(SumOfDivisors(n),n)): n in [1..100]]; // _Vincenzo Librandi_, Oct 09 2017
%Y A073802 Cf. A000005, A000203, A062068, A073803, A073804.
%K A073802 nonn
%O A073802 1,6
%A A073802 _Labos Elemer_, Aug 13 2002