well known sieve?

Question

Incidentally I've shown the following fact.

Cancelling from the natural numbers $1$, the even integers, the integers of the form $n^2+2nk$, with $n>1$ odd and $k$ nonnegative, we obtain all (and only) the prime numbers.

Is it well known?

Answer 1

Yes; your result is that every odd composite integer $N$ can be written in the form $N=n(n+2k)$ for some odd $n$ and $k \geq 0$. This follows immediately by considering the smallest divisor $n>1$ of $N$ and noting that $N/n$ is odd and not smaller than $n$. Of course, primes cannot be written in this form.