Wolfram MathWorld says:
An infinite number of weird numbers are known to exist, and the sequence of weird numbers has positive Schnirelmann density.
Wikipedia says:
Infinitely many weird numbers exist.$^{[3]}$ For example, $70p$ is weird for all primes $p ≥ 149$. In fact, the set of weird numbers has positive asymptotic density.$^{[4]}$
Asymptotic density differs from Schnirelmann density, and so I was wondering; which density does the set of weird numbers actually have? Could it have both types? I find it a bit odd that each page would list a different type of density.
I could not find a reference for what Wolfram says, so I am a bit confused about where it comes from. In this previous question of mine, I asked how Erdos arrived at a positive density. His paper says "we prove that the density of weird numbers is positive", however, it does not explain which type of density they have. This is the same paper that Wikipedia cites in $[4]$, so I am wondering, could Wolfram Mathworld have it wrong?
By the way, weird numbers are defined as the following:
An integer n is called weird if n is abundant but not pseudoperfect .
Thank you.
It seems that MathWorld has it wrong: The smallest weird number is $70$ (as MathWorld agrees), hence $$\sigma A=\inf_{n\in\Bbb N}\frac{|A\cap\{1,\ldots,n\}|}n\le \frac{|A\cap\{1,\ldots,42\}|}{42}=0 $$ holds for the set $A$ of weird numbers.