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A103339
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Numerator 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, 4, 3, 8, 5, 2, 7, 16, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 8, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 80, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 28, 57, 116, 59, 4, 61, 31, 63, 128, 65, 11, 67, 136, 23, 35, 71, 16, 73
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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) = 16 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(numer(uH(n)), n=1..81);
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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_] := Numerator[n*ud[n]/usigma[n]]; Array[a, 100] (* Jean-François Alcover, Dec 03 2016 *)
a[n_] := Numerator[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 ((%), numerator)
a103339 = numerator . uhm where uhm n = (n * a034444 n) % (a034448 n)
(Python)
from sympy import gcd
from sympy.ntheory.factor_ import udivisor_sigma
def A103339(n): return (lambda x, y: y*n//gcd(x, y*n))(udivisor_sigma(n), udivisor_sigma(n, 0)) # Chai Wah Wu, Oct 20 2021
(PARI) a(n) = {my(f = factor(n)); numerator(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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