Perfection quotient ($\sigma (n)/n$)

The perfection quotient of a positive integer is the ratio $\sigma (n)/n$, where $\sigma$ is the sum-of-divisors function. It distinguishes the three familiar classes: a deficient number has quotient below two, a perfect number has quotient two, and an abundant number has quotient above two. For example, the quotients of $8$, $6$ and $12$ are $15/8$, $2$ and $7/3$.

For coprime integers $a,b$, the quotient is multiplicative, since $\sigma (ab)/(ab)=(\sigma (a)/a)(\sigma (b)/b)$. At a prime power, its factor isConsequently a prescribed quotient can be studied one prime power at a time, tracking the remaining numerator and denominator.