A103340 Denominator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

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

Offset: 1

Author

Emeric Deutsch, Jan 31 2005

Keywords

• frac
• nonn

Examples

			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.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A103339 (numerators), A099377, A099378.

Cf. A034444, A034448, A077610.

Programs

Haskell

import Data.Ratio ((%), denominator)
a103340 = denominator . uhm where uhm n = (n * a034444 n) % (a034448 n)
-- Reinhard Zumkeller, Mar 17 2012

Maple

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);

Mathematica

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 *)

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

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

Formula

a(A006086(n)) = 1. - Reinhard Zumkeller, Mar 17 2012