Are weird numbers more rare than prime numbers?

Question

By taking a look at the first few weird numbers: $$(70, 836, 4030, 5830, 7192, 7912, 9272, 10430)$$ It is certain that prime numbers occurs more often within this range of numbers.

But are weird numbers more rare than prime numbers in the long run? Sure, by the definition of infinity, there are infinite prime numbers and infinite weird numbers. But if you calculated prime numbers and weird numbers for a finite amount of time, would prime numbers be more common than weird numbers?

This may not be very easy to explain, but I'd appreciate an attempt to keep it as simple as possible.

Answer 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 $\pi(n)\sim\frac{n}{\log n}$.

Answer 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.