If my calculations with my computer were rights, when I've consider the sequence of integers $n\geq 1$ satisfying $$\sigma(n)\mid (n(\sigma_0(n))^2),$$ where $\sigma(n)=\sum_{d\mid n}d$ is the sum of divisors function and $\sigma_0(n)=\sum_{d\mid n}1$ the number of such divisors, then such sequence starts as
1, 3, 6, 15, 28, 33, 42, 84, 91...
but I believe that such sequence isn't in The On-Line Encyclopedia of Integer Sequence.
Question. Are right my first terms? Is known such sequence in previous encyclopedia or are there references in the literature?
I've curiosity to know what's about the number of terms in the sequence for the first $10^k$ with $k=4,5,6,\cdots, N$, thats is a plot of the counting function for previous sequence.
Question. Can you provide us a graph of the counting function of such sequence for $10^N$, for a $N$ large? Thanks in advance.
OEIS refer to section B2 in Guy, Unsolved Problems in Number Theory, for types of "semi" perfect numbers. They do define the harmonic numbers but do not say much more. Anyway, your sequence is a superset of the harmonic numbers (named by Pomerance in 1973), also called Ore numbers.
The harmonic numbers begin $$ 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, $$ and are discussed at https://oeis.org/A001599