I know that the definition of a perfect number is that its divisor function value is equal to double the number, i.e.,
$$\sigma(n) = 2n$$
By accident I came across the number $250801742479451287$, which divisor function value is almost equal to itself, i.e.,
$$\frac{\sigma(250801742479451287)}{250801742479451287}=\frac{250824161834757504}{250801742479451287}=1.00008939...$$
By playing around, it seems that this might not be some special case, and probably in general, the higher the $n$, the more often and more closer to $1$ we happen to be with the ratio $\frac{\sigma(n)}{n}$.
My wording might be not correct, but is there something known about the asymptotics of $\sigma(n)$ or even better about the ratio $\frac{\sigma(n)}{n}$? Is it bounded? Does it converge? If so, does it mean that the higher the $n$, the less likely are the perfect numbers (multi-perfect numbers)? Thanks.