Are weird numbers more rare than prime numbers?

Solution 1:

Wikipedia cites Benkoski, Stan; Erdős, Paul (April 1974). "On Weird and Pseudoperfect Numbers" for the fact the weird numbers have positive asymptotic density. But primes have zero asymptotic density, so in a sense, in a long run weird numbers are not only more abundant, but infinitely more abundant. More quantitatively, if we let $w(n)$ be the weird-number-counting function, we should have $w(n)\sim \alpha n$ for some parameter $0<\alpha<1$, whereas the prime number theorem tells us $\pi(n)\sim\frac{n}{\log n}$.

Solution 2:

Wanted to just leave this as a comment but this is probably easier. As you probably know the OEIS usually has an abudance of information for things like this , https://oeis.org/A006037 , just by inspection one can see that the wierd numbers be come more dense as they grow in size. Far from a proof but useful in getting an idea of their denisty.