harmonic divisor | Ore number

A number n for which the harmonic mean of the divisors of n, i.e., n d(n)/σ(n), is an integer, where d(n) = σ_0(n) is the number of positive integer divisors of n and σ(n) = σ_1(n) is the divisor function. For example, the divisors of n = 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140, giving d(140) | = | 12 σ(140) | = | 336 (140d(140))/(σ(140)) | = | (140·12)/336 = 5, so 140 is a harmonic divisor number. Harmonic divisor numbers are also called Ore numbers. Garcia gives the 45 harmonic divisor numbers less than 10^7. The first few are 1, 6, 28, 140, 270, 496, ... (OEIS A001599).

divisor function | harmonic number