Eccoci.

For an integer $n \ge 4$, we define the lower prime square root of $n$, denoted by $\operatorname{lps}(n)$, as the largest prime $\le \sqrt n$ and the upper prime square root of $n$, $\operatorname{ups}(n)$, as the smallest prime $\ge \sqrt n$.

So, for example, $\operatorname{lps}(4) = 2 = \operatorname{ups}(4)$, $\operatorname{lps}(1000) = 31$, $\operatorname{ups}(1000) = 37$.
Let us call an integer $n \ge 4$ semidivisible, if one of $\operatorname{lps}(n)$ and $\operatorname{ups}(n)$ divides $n$, but not both.

The sum of the semidivisible numbers not exceeding $15$ is $30$, the numbers are $8$, $10$ and $12$.
$15$ is not semidivisible because it is a multiple of both $\operatorname{lps}(15) = 3$ and $\operatorname{ups}(15) = 5$.
As a further example, the sum of the $92$ semidivisible numbers up to $1000$ is $34825$.

What is the sum of all semidivisible numbers not exceeding $999966663333$?