A non-square number cannot be factored to two identical factors. However, not all non-squares are equal: some of them can be factored to relatively close factors (for example, $6=2*3$), while others can be factored only to faraway factors (for example, $27=9*3$).
Formally, define a number $N$ as $R$-balanced (for $R \geq 1$), if it can be factored as $N=A*B$, such that $A \geq B$, and $A / B \leq R$. For example:
- $9=3*3$, therefore (like all other square numbers) is $1$-balanced, and also $2$-balanced, $3$-balanced, etc.
- $6=2*3$, therefore it is $1.5$-balanced, and also $2$-balanced, but not $1$-balanced.
- $27=9*3$, therefore it is $3$-balanced.
- $7=7*1$, therefore it is $7$-balanced (similarly, every prime number $p$ is $p$-balanced).
I am interested in the density of the balanced integers. Specifically:
A. Is there an asymptotic formula for the number of $R$-balanced integers smaller than $N$?
- For $R=1$, it is easy - the number of $1$-balanced integers smaller than $N$ is $\theta (\sqrt{N})$, so the density is $\theta (1/\sqrt{N})$, which goes to $0$ as $N$ goes to infinity. Is this also true for $R=2$? Is there an $R$ for which the density of $R$-balanced numbers becomes constant?
B. Is there an asymptotic formula for the distance between an integer $N$ and the closest $R$-balanced integer larger than $N$?
- For $R=1$, it is easy - the closest $1$-balanced integer larger than $N$ is $\lceil\sqrt{N}\rceil^2$, and the distance from $N$ is at most $2 \sqrt{N} + 1$, which goes to infinity as $N$ goes to infinity. Is this also true for $R=2$? Is there an $R$ for which the distance to the closest $R$-balanced number becomes constant?