On Exactly $3$-Deficient-Perfect Numbers

Abstract

Let $n$ and $k$ be positive integers and $\sigma (n)$ the sum of all positive divisors of $n$. We call $n$ an exactly $k$-deficient-perfect number with deficient divisors $d_1,d_2,\dots ,d_k$ if $d_1,d_2,\dots ,d_k$ are distinct proper divisors of $n$ and $\sigma (n)=2n-(d_1+d_2+\dots +d_k)$. In this article, we show that the only odd exactly $3$-deficient-perfect number with at most two distinct prime factors is $1521=3^2\cdot 13^2$.

submitted