8 problems

Let $n=2^k p^m$ be 2-near perfect, where $p$ is an odd prime and $m\geq 3$. The paper proves that then $m=3$, $p=2^{k+1}-1$, and the omitted divisors are $p$ and $p^2$. Mersenne-pr…

Let $n=2^k p^m$, where $p$ is an odd prime, and suppose that $n$ is 2-near perfect when … for distinct positive divisors $d_1,d_2$ of $n$. Finiteness conjecture. There are only fin…

Let a Kalita–Saikia near superperfect number mean a number that is both a Kalita–Saikia number and a near superperfect number. Nonexistence conjecture. There are no odd Kalita–Saik…

Fix an integer $m\geq2$, and consider 2-near perfect numbers of the form $2^kp^m$, where $k$ is an integer and $p$ is prime. Fixed-exponent finiteness conjecture. For any fixed…

Let a 2-near perfect number be a positive integer with the relevant divisor-sum property, and consider numbers of the form $2^kp^m$, where $k$ and $m$ are integers and $p$ is prime…

A positive integer $n$ is near-perfect if it is the sum of all its proper divisors except one proper divisor, called its redundant divisor. Evenness conjecture. Every near-perfect…

A positive integer $n$ is near-perfect if it is the sum of all its proper divisors except one proper divisor, called its redundant divisor. Classification conjecture. Every near-pe…

A positive integer $n$ is near-perfect if it is the sum of all its proper divisors except one proper divisor, called its redundant divisor. For a fixed integer $k\geq 1$, the redun…