A103339 - OEIS

A103339

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

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

1,2

LINKS

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

FORMULA

a(A006086(n)) = A006087(n). - Reinhard Zumkeller, Mar 17 2012

From Amiram Eldar, Mar 10 2023: (Start)

a(n)/A103340(n) = n*A034444(n)/A034448(n).

a(n)/A103340(n) <= A099377(n)/A099378(n), with equality if and only if n is squarefree (A005117). (End)

EXAMPLE

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.

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(numer(uH(n)), n=1..81);

MATHEMATICA

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 *)

PROG

(Haskell)

import Data.Ratio ((%), numerator)

a103339 = numerator . uhm where uhm n = (n * a034444 n) % (a034448 n)

-- Reinhard Zumkeller, Mar 17 2012

(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

CROSSREFS