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A099378 Denominators of the harmonic means of the divisors of the positive integers.

%I A099378 #24 Feb 16 2025 08:32:55
%S A099378 1,3,2,7,3,1,4,15,13,9,6,7,7,3,2,31,9,13,10,7,8,9,12,5,31,21,10,1,15,
%T A099378 3,16,21,4,27,12,91,19,15,14,9,21,2,22,7,13,9,24,31,19,31,6,49,27,5,
%U A099378 18,15,20,45,30,7,31,12,52,127,21,3,34,21,8,9,36,65,37,57,62,35,24,7,40,93
%N A099378 Denominators of the harmonic means of the divisors of the positive integers.
%H A099378 Ivan Neretin, <a href="/A099378/b099378.txt">Table of n, a(n) for n = 1..10000</a>
%H A099378 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OresConjecture.html">Ore's Conjecture</a>
%e A099378 1, 4/3, 3/2, 12/7, 5/3, 2, 7/4, 32/15, ...
%t A099378 f[n_] := DivisorSigma[0, n]/Plus @@ (1/Divisors@n); Denominator@ Array[f, 80] (* _Robert G. Wilson v_, Aug 04 2010 *)
%t A099378 Table[Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]], {n, 80}] (* _Ivan Neretin_, Nov 13 2016 *)
%o A099378 (PARI) a(n) = my(d=divisors(n)); denominator(#d/sum(k=1, #d, 1/d[k])); \\ _Michel Marcus_, Nov 13 2016
%o A099378 (PARI) first(n)=my(v=vector(n)); forfactored(k=1,n, v[k[1]]=denominator(sigma(k, 0)/sigma(k, -1))); v \\ _Charles R Greathouse IV_, Nov 01 2021
%o A099378 (Python)
%o A099378 from sympy import gcd, divisor_sigma
%o A099378 def A099378(n): return (lambda x, y: x//gcd(x,y*n))(divisor_sigma(n),divisor_sigma(n,0)) # _Chai Wah Wu_, Oct 20 2021
%Y A099378 Cf. A099377.
%K A099378 nonn,frac
%O A099378 1,2
%A A099378 _Eric W. Weisstein_, Oct 13 2004
%E A099378 More terms from _Robert G. Wilson v_, Aug 04 2010