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 A073802 Number of common divisors of n and sigma(n). 16
 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 (list; graph; refs; listen; history; text; internal format)
 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}, NCD[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 *) 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 * A132157 A103163 A128211 Adjacent sequences: A073799 A073800 A073801 * A073803 A073804 A073805 KEYWORD nonn AUTHOR Labos Elemer, Aug 13 2002 STATUS approved

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Last modified February 5 11:26 EST 2023. Contains 360084 sequences. (Running on oeis4.)