I guess you mean $15$ proper divisors; it's less confusing to say $16$ divisors, and counting all the divisors has the nicest formula.

The sequence of weird numbers is A006037 on the OEIS; looking at some of their factorizations it looks like $16 = 2^4$ and these numbers are products of four distinct primes, e.g.

$$19390 = 2 \cdot 5 \cdot 7 \cdot 277, 18970 = 2 \cdot 5 \cdot 7 \cdot 271, , 18830 = 2 \cdot 5 \cdot 7 \cdot 269.$$

Actually these factorizations suggests a much more specific conjecture: that we are specifically seeing numbers of the form $70p$ where $p$ is a large prime. $70$ is the smallest weird number, and on the OEIS you can find the following remark:

A weird number $n$ multiplied by a prime $p > \sigma(n)$ is again weird.

So any number of the form $70p$ where $p > \sigma(70) = 144$ is prime is weird and I'll bet that's where most of your weird numbers with $16$ divisors are coming from. These have asymptotic density $\frac{1}{70 \ln \frac{n}{70}}$ among the numbers from $1$ to $n$ by the PNT.

The next smallest weird number is $836 = 2^2 \cdot 11 \cdot 19$ so we might guess that the next most common weird numbers are of the form $836p$ where $p > \sigma(836) = 1680$ is prime. These will have $24$ divisors which is the next most common number in your count. These have asymptotic density $\frac{1}{836 \ln \frac{n}{836}}$ which is about a tenth of the density of the previous, which is roughly consistent with your count.

A002975 records the primitive weird numbers, namely those not divisible by a previous weird number. These exhibit a different pattern of factorizations; the largest one listed on the OEIS is $4128448 = 2^6 \times 251 \times 257$ which has $28$ divisors, for example. So it might be interesting to restrict your attention to primitive weird numbers and retest.

I guess you mean $15$ proper divisors; it's less confusing to say $16$ divisors, and counting all the divisors has the nicest formula.

The sequence of weird numbers is A006037 on the OEIS; looking at some of their factorizations it looks like $16 = 2^4$ and these numbers are products of four distinct primes, e.g.

$$19390 = 2 \cdot 5 \cdot 7 \cdot 277, 18970 = 2 \cdot 5 \cdot 7 \cdot 271, , 18830 = 2 \cdot 5 \cdot 7 \cdot 269.$$

Actually these factorizations suggests a much more specific conjecture: that we are specifically seeing numbers of the form $70p$ where $p$ is a large prime. $70$ is the smallest weird number, and on the OEIS you can find the following remark:

A weird number $n$ multiplied by a prime $p > \sigma(n)$ is again weird.

So any number of the form $70p$ where $p > \sigma(70) = 144$ is prime is weird and I'll bet that's where most of your weird numbers with $16$ divisors are coming from. These have asymptotic density $\frac{1}{70 \ln \frac{n}{70}}$ among the numbers from $1$ to $n$ by the PNT.

The next smallest weird number is $836 = 2^2 \cdot 11 \cdot 19$ so we might guess that the next most common weird numbers are of the form $836p$ where $p > \sigma(836) = 1680$ is prime. These will have $24$ divisors which is the next most common number in your count. These have asymptotic density $\frac{1}{836 \ln \frac{n}{836}}$ which is about a tenth of the density of the previous, which is roughly consistent with your count.

The next smallest weird number is $4030 = 2 \cdot 5 \cdot 13 \cdot 31$ and from this we get weird numbers of the form $4030p$ where $p > \sigma(4030) = 8064$ with $32$ divisors. But this gives $4030p > 3.2 \times 10^7$ which is past where you searched.

A002975 records the primitive weird numbers, namely those not divisible by a previous weird number. These exhibit a different pattern of factorizations; the largest one listed on the OEIS is $4128448 = 2^6 \times 251 \times 257$ which has $28$ divisors, for example. So it might be interesting to restrict your attention to primitive weird numbers and see what the counts are in that case. Instead of just computing and comparing the number of divisors it would be more illuminating to compute and compare the sequence of exponents in the prime factorization.

I guess you mean $15$ proper divisors; it's less confusing to say $16$ divisors, and counting all the divisors has the nicest formula.

The sequence of weird numbers is A006037 on the OEIS; looking at some of their factorizations it looks like $16 = 2^4$ and these numbers are products of four distinct primes, e.g.

$$19390 = 2 \cdot 5 \cdot 7 \cdot 277, 18970 = 2 \cdot 5 \cdot 7 \cdot 271, , 18830 = 2 \cdot 5 \cdot 7 \cdot 269.$$

Actually these factorizations suggests a much more specific conjecture: that we are specifically seeing numbers of the form $70p$ where $p$ is a large prime. $70$ is the smallest weird number, and on the OEIS you can find the following remark:

A weird number $n$ multiplied by a prime $p > \sigma(n)$ is again weird.

So any number of the form $70p$ where $p > \sigma(70) = 144$ is prime is weird and I'll bet that's where most of your weird numbers with $16$ factors are coming from. These have asymptotic density $\frac{1}{70 \ln n}$ among the numbers from $1$ to $n$ by the PNT.

The next smallest weird number is $836 = 2^2 \cdot 11 \cdot 19$ so we might guess that the next most common weird numbers are of the form $836p$ where $p > \sigma(836) = 1680$ is prime. These will have $24$ divisors which is the next most common number in your count. These have asymptotic density $\frac{1}{836 \ln n}$ which is a little less than a tenth of the density of the previous, which is roughly consistent with your count.

A002975 records the primitive weird numbers, namely those not divisible by a previous weird number. These exhibit a different pattern of factorizations; the largest one listed on the OEIS is $4128448 = 2^6 \times 251 \times 257$ which has $28$ divisors, for example.

I guess you mean $15$ proper divisors; it's less confusing to say $16$ divisors, and counting all the divisors has the nicest formula.

The sequence of weird numbers is A006037 on the OEIS; looking at some of their factorizations it looks like $16 = 2^4$ and these numbers are products of four distinct primes, e.g.

$$19390 = 2 \cdot 5 \cdot 7 \cdot 277, 18970 = 2 \cdot 5 \cdot 7 \cdot 271, , 18830 = 2 \cdot 5 \cdot 7 \cdot 269.$$

Actually these factorizations suggests a much more specific conjecture: that we are specifically seeing numbers of the form $70p$ where $p$ is a large prime. $70$ is the smallest weird number, and on the OEIS you can find the following remark:

A weird number $n$ multiplied by a prime $p > \sigma(n)$ is again weird.

So any number of the form $70p$ where $p > \sigma(70) = 144$ is prime is weird and I'll bet that's where most of your weird numbers with $16$ divisors are coming from. These have asymptotic density $\frac{1}{70 \ln \frac{n}{70}}$ among the numbers from $1$ to $n$ by the PNT.

The next smallest weird number is $836 = 2^2 \cdot 11 \cdot 19$ so we might guess that the next most common weird numbers are of the form $836p$ where $p > \sigma(836) = 1680$ is prime. These will have $24$ divisors which is the next most common number in your count. These have asymptotic density $\frac{1}{836 \ln \frac{n}{836}}$ which is about a tenth of the density of the previous, which is roughly consistent with your count.

A002975 records the primitive weird numbers, namely those not divisible by a previous weird number. These exhibit a different pattern of factorizations; the largest one listed on the OEIS is $4128448 = 2^6 \times 251 \times 257$ which has $28$ divisors, for example. So it might be interesting to restrict your attention to primitive weird numbers and retest.