The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le 10^8$ the ratio $n/v_n$ is different for each $n$.
Unique variance conjecture: The ratio of a number to the variance of its divisors is injective.
Question: Is there any explanation why this should be true?
Note: This question was posted in MSE a week ago but did not get an answer
Related questions: Does the average primeness of natural numbers tend to zero?.
