On odd deficient-perfect numbers with four distinct prime divisors

Abstract

For a positive integer $n$, let $\sigma (n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $\sigma (n)=2n-d$. In this paper, we show that the only odd deficient-perfect number with four distinct prime divisors is $3^2\cdot 7^2\cdot 11^2\cdot 13^2$.

47 pages