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

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, 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, 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

Offset: 1

Author

Eric W. Weisstein, Oct 13 2004

Keywords

• nonn
• frac

Examples

			1, 4/3, 3/2, 12/7, 5/3, 2, 7/4, 32/15, ...
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Ore's Conjecture

Crossrefs

Cf. A099377.

Programs

Mathematica

f[n_] := DivisorSigma[0, n]/Plus @@ (1/Divisors@n); Denominator@ Array[f, 80] (* Robert G. Wilson v, Aug 04 2010 *)
Table[Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]], {n, 80}] (* Ivan Neretin, Nov 13 2016 *)

PARI

a(n) = my(d=divisors(n)); denominator(#d/sum(k=1, #d, 1/d[k])); \\ Michel Marcus, Nov 13 2016

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

Python

from sympy import gcd, divisor_sigma
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

Extensions

More terms from Robert G. Wilson v, Aug 04 2010