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A103340
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Denominator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.
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10
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1, 3, 2, 5, 3, 1, 4, 9, 5, 9, 6, 5, 7, 3, 2, 17, 9, 5, 10, 3, 8, 9, 12, 3, 13, 21, 14, 5, 15, 3, 16, 33, 4, 27, 12, 25, 19, 15, 14, 27, 21, 2, 22, 15, 1, 9, 24, 17, 25, 39, 6, 35, 27, 7, 18, 9, 20, 45, 30, 1, 31, 12, 20, 65, 21, 3, 34, 45, 8, 9, 36, 5, 37, 57, 26, 25, 24, 7, 40, 51, 41, 63
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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1, 4/3, 3/2, 8/5, 5/3, 2, ...
a(8) = 9 because the unitary divisors of 8 are {1,8} and 2/(1/1 + 1/8) = 16/9.
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MAPLE
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with(numtheory): udivisors:=proc(n) local A, k: A:={}: for k from 1 to tau(n) do if gcd(divisors(n)[k], n/divisors(n)[k])=1 then A:=A union {divisors(n)[k]} else A:=A fi od end: utau:=n->nops(udivisors(n)): usigma:=n->sum(udivisors(n)[j], j=1..nops(udivisors(n))): uH:=n->n*utau(n)/usigma(n):seq(denom(uH(n)), n=1..90);
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MATHEMATICA
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ud[n_] := 2^PrimeNu[n]; usigma[n_] := DivisorSum[n, If[GCD[#, n/#] == 1, #, 0]&]; a[1] = 1; a[n_] := Denominator[n*ud[n]/usigma[n]]; Array[a, 100] (* Jean-François Alcover, Dec 03 2016 *)
a[n_] := Denominator[n * Times @@ (2 / (1 + Power @@@ FactorInteger[n]))]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 10 2023 *)
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PROG
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(Haskell)
import Data.Ratio ((%), denominator)
a103340 = denominator . uhm where uhm n = (n * a034444 n) % (a034448 n)
(Python)
from sympy import gcd
from sympy.ntheory.factor_ import udivisor_sigma
def A103340(n): return (lambda x, y: x//gcd(x, y*n))(udivisor_sigma(n), udivisor_sigma(n, 0)) # Chai Wah Wu, Oct 20 2021
(PARI)
a(n) = {my(f = factor(n)); denominator(n * prod(i=1, #f~, 2/(1 + f[i, 1]^f[i, 2]))); } \\ Amiram Eldar, Mar 10 2023
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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