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A073802
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Number of common divisors of n and sigma(n).
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16
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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
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OFFSET
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1,6
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COMMENTS
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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)
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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FORMULA
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See program.
a(n) = A000005(A009194(n)) = tau(gcd(n,sigma(n))). [Reinhard Zumkeller, Mar 12 2010]
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000005, A000203, A062068, A073803, A073804.
Sequence in context: A269443 A039927 A336722 * A132157 A103163 A128211
Adjacent sequences: A073799 A073800 A073801 * A073803 A073804 A073805
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KEYWORD
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nonn
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AUTHOR
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Labos Elemer, Aug 13 2002
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STATUS
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approved
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