Why not generalize the concept of prime numbers?

Question

A prime number is a natural number N > 1, that is divisible only by 1 and N (N/1).
The number 1 is given special treatment in this definition, and this can be generalized by extending the set of the specially treated numbers, for example to also include number 2, lets call these secondary numbers:
A secondary number is a natural number N > 2, that is divisible only by 1, N/1, 2, N/2.
This gives us a completely new set of 'prime numbers' that have many qualities of prime number set,
theres infinitely many of them and it seems the fundamental theorem of arithmetics is also true for this set.
Would this set be somehow less special than that of prime numbers?
Did anyone already do this?

lhf · Answer

Here is one natural generalization along these lines:

A natural number $n$ is $k$-prime when it has exactly $k+1$ positive divisors.

Then $1$-prime is the same as ordinary prime. And $0$-prime is the unit $1$. Also, a $2$-prime is the square of a prime but a $3$-prime is the cube of a prime or the product of two primes.
The problem with this generalization is that $k$-primes do not occur naturally in number theory and so are not very useful.

Why not generalize the concept of prime numbers?

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