On Exactly $3$-Deficient-Perfect Numbers

On odd deficient-perfect numbers with four distinct prime divisors

For a positive integer $n$, let $σ(n)$ 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 $σ(n)=2n-d$. In this paper, we show that the only odd deficient-perfect number with four distinct prime divisors is $3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}$.

On Deficient Perfect Numbers with Four Distinct Prime Factors

For a positive integer $n$, if $σ(n)$ denotes the sum of the positive divisors of $n$, then $n$ is called a deficient perfect number if $σ(n)=2n-d$ for some positive divisor $d$ of $n$. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.

奇完全数或奇本原非亏数素因子倒数之和

The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number

本文在奇完全数及更弱的奇本原非亏数假设下，证明了素因子倒数之和 T(n) 与乘积 H(n) 关于最小素因子的新上界。

On the small prime factors of a non-deficient number

Let $σ(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $σ(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. We also obtain tighter bounds for odd perfect numbers. We also discuss the behavior of $σ(n!+1)$, $σ(2^n+1)$, and related sequences.

A Criterion for Deficient Numbers Using the Abundancy Index and Deficiency Functions

We show that $n$ is almost perfect if and only if $I(n) - 1 < D(n) \leq I(n)$, where $I(n)$ is the abundancy index of $n$ and $D(n)$ is the deficiency of $n$. This criterion is then extended to the case of integers $m$ satisfying $D(m)>1$.

恰好有两个素因子的 $2$-近完美数的完整刻画

Complete characterization of $2$-near perfect numbers with exactly 2 prime factors

本文通过分析奇偶性及指数条件，完整刻画了恰好有两个素因子的 $2$-近完美数，证明不存在奇数的此类数，并确定了所有偶数形式属于特定族。