Numbers that are NOT prime powers.

Question

Is the set of numbers that are NOT prime powers infinite?

I know there is an infinite amount of primes...and therefore an infinite amount of prime powers.

I'm just curious if the set if those that are nit prime powers is also infinite.

Thanks!

Answer 1

Of course the set of numbers that are not prime powers is infinite.

If I'm understanding your question correctly, what you're looking for are the squarefree composite numbers. Ten thousand of them are listed in Sloane's A120944, and the only reason it doesn't list more is because the page would take too long to load.

There are several ways you could filter that set and still have an infinite set. For instance, you could decide you just want numbers that are the product of three distinct primes. In a finite list of consecutive integers, there probably are fewer of those than there are squarefree numbers with at least two prime factors, but among all integers, both sets are infinite.