Harmonic Numbers

On January 1st 2024, I made a post titled Unitary Harmonic Numbers which are defined as numbers whose unitary divisors have a harmonic mean that is an integer. Oddly, I have never made a post simply about harmonic numbers defined as numbers whose divisors have a harmonic mean that is an integer. Like unitary harmonic numbers, they are quite rare. The number associated with my diurnal age today (27846) is one such harmonic number. These numbers make up OEIS A001599 and the initial members up to one million are (perfect numbers are shown in red):

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976

Here is what Numbers Aplenty had to say about them:

A number $n$  is called harmonic divisor number if the harmonic mean of its divisors is an integer. This is equivalent to saying that the average of the divisors of $n$ divides $n$: $$\frac{n}{\sigma (n)/\tau (n)}=\frac{n\times \tau (n)}{\sigma (n)}\text{ is an integer}$$Harmonic divisor numbers are also called harmonic numbers, for brevity, or Ore numbers, after O.Ore who studied them. He proved that all the perfect numbers are also harmonic and conjectured that 1 is the only odd harmonic number. This conjecture has been verified by G.L.Cohen et al. for $n<{10}^{24}$ and if true, it will imply that no odd perfect numbers exist. Jaycob Coleman has observed that all the Ore numbers up to ${10}^{14}$ are also practical numbers and conjectured this holds in general. T. Goto and K. Okeya have computed a list of the 937 harmonic numbers up to ${10}^{14}$.

In the case of the number (27846) associated with my diurnal age, we have:$$\begin{array}{rl}\tau (27846)& =48\\ \sigma (27846)& =78624\\ \frac{27846\times 48}{78624}& =17\end{array}$$