Is every multiple of a perfect number a semiperfect number? If not, when isn't it?
A perfect number is a number that is equal to the sum of its proper divisors (divisors not including the number itself but including 1.)
A semiperfect number is a number that is equal to the sum of some of its proper divisors (divisors not including the number itself but including 1.)
Let $n$ be a perfect number and let $d_1,\dots,d_k$ be its proper divisors, so that $d_1+\cdots+d_k=n$. For any multiple $mn$ of $n$, each of $md_1,\dots,md_k$ is a proper divisor of $mn$, and $md_1+\cdots+md_k=mn$; hence $mn$ satisfies your definition of semiperfect.