Mathematicians have very precise definitions for terms like "infinite" and "same size". The single unambiguous correct answer to this question is that using the standard mathematical definitions, the rationals have the "same size" as the integers.
First, here are the definitions:
Define "$0$" = emptyset, "$1$" $= \{0\}$, "$2$" $= \{0,1\}$, "$3$" $= \{0,1,2\}$, etc. So, the number "n" is really a set with $n$ elements in it.
A set $A$ is called "finite" iff there is some $n$ and a function $f:A\to n$ which is bijective.
A set $A$ is called "infinite" iff it is not finite. (Note that this notion says nothing about "counting never stops" or anything like that.)
Two sets $A$ and $B$ are said to have the "same size" if there is a function $f:A\to B$ which is a bijection. Note that we do NOT require that ALL functions be bijections, just that there is SOME bijection.
Once one accepts these definitions, one can prove that the rationals and integers have the same size. One just needs to find a particular bijection between the two sets. If you don't like the one you mentioned in your post, may I suggest the Calkin-Wilf enumeration of the rationals?
Of course, these give bijections between the naturals (without $0$) and the rationals, but once you have a bijection like this, it's easy to construct a bijection from the integers to the rationals by composing with a bijection from the integers to the positive natural numbers.
I think that the issue here is mostly linguistic.
Saying that something is infinite simply says that it is not finite. Saying that I have more than two students in my class tells you nothing about whether there are three, or six, or 42 students in my class.
It is true that both $\Bbb N$ and $\Bbb R$ are infinite. But it tells you nothing about comparing their sizes in a meaningful way.
There are several ways to measure the size of a set, depending on the context. In the case of subsets of $\Bbb R$ we can ask what is their length, or how can we approximate their size using things which have length (namely, intervals). This gives us measure theory, and it turns out that under some reasonable assumptions we cannot assign "length" to every set of real numbers.
The same can be done in $\Bbb R^2$, where now length is replaced by area, or in $\Bbb R^3$ with volume, and so on.
If you want only to consider sets of natural numbers there are ways to measure the size of those sets, in a fashion that assigns bigger and smaller sets some notion of largeness that fits some sort of intuition.
But those are just sets of real numbers, or subsets of the space, or something like that. What about much larger sets? Like the set of all sets of reals? Or all sets of reals which have "length" assigned to them, or so on? What about sets of those sets, or sets of sets of sets of sets of those... etc.
At some point, all the nice structure that the real numbers and related objects carry with them goes away and disappears. Bijections, they stay with you forever. So we measure the size of sets using functions which have certain properties, namely they are injective and surjective.
We say that two sets have the same size, or same cardinality if there is a bijection between them. This bijection does not need to preserve any given structure. The natural numbers look nothing like the rational numbers, but both are countably infinite, for example.
And as it turns out, there is a bijection between $[0,1]$ and $[0,2]$ as intervals of the real line; but there is no bijection between $\Bbb N$ and $\Bbb R$. Since there is an injection from $\Bbb N$ to $\Bbb R$, this means that there cardinality of $\Bbb N$ is strictly smaller than that of $\Bbb R$. Cantor's theorem also tells us that if $X$ is any given set, then $\mathcal P(X)$ which is the set of all subsets of $X$, has a larger cardinality than $X$. So any set has more subsets than it has elements. Even infinite ones.
Best Answer
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 $\pi(n)\sim\frac{n}{\log n}$.