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A103339 Numerator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

%I A103339 #29 Mar 10 2023 09:10:01
%S A103339 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,
%T A103339 14,29,10,31,64,11,68,35,72,37,38,39,80,41,7,43,44,3,23,47,48,49,100,
%U A103339 17,104,53,18,55,28,57,116,59,4,61,31,63,128,65,11,67,136,23,35,71,16,73
%N A103339 Numerator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.
%H A103339 Reinhard Zumkeller, <a href="/A103339/b103339.txt">Table of n, a(n) for n = 1..10000</a>
%F A103339 a(A006086(n)) = A006087(n). - _Reinhard Zumkeller_, Mar 17 2012
%F A103339 From _Amiram Eldar_, Mar 10 2023: (Start)
%F A103339 a(n)/A103340(n) = n*A034444(n)/A034448(n).
%F A103339 a(n)/A103340(n) <= A099377(n)/A099378(n), with equality if and only if n is squarefree (A005117). (End)
%e A103339 1, 4/3, 3/2, 8/5, 5/3, 2, ...
%e A103339 a(8) = 16 because the unitary divisors of 8 are {1,8} and 2/(1/1 + 1/8) = 16/9.
%p A103339 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);
%t A103339 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 *)
%t A103339 a[n_] := Numerator[n * Times @@ (2 / (1 + Power @@@ FactorInteger[n]))]; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Mar 10 2023 *)
%o A103339 (Haskell)
%o A103339 import Data.Ratio ((%), numerator)
%o A103339 a103339 = numerator . uhm where uhm n = (n * a034444 n) % (a034448 n)
%o A103339 -- _Reinhard Zumkeller_, Mar 17 2012
%o A103339 (Python)
%o A103339 from sympy import gcd
%o A103339 from sympy.ntheory.factor_ import udivisor_sigma
%o A103339 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
%o A103339 (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
%Y A103339 Cf. A103340 (denominators), A099377, A099378.
%Y A103339 Cf. A005117, A034444, A034448, A077610.
%K A103339 frac,nonn
%O A103339 1,2
%A A103339 _Emeric Deutsch_, Jan 31 2005