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In this paper, we prove that, if 
$N$ is a positive odd number with 
$r$ distinct prime factors such that 
$N\mid \sigma (N)$, then 
$N\lt {2}^{{4}^{r} - {2}^{r} } $ and 
$N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where 
$\sigma (N)$ is the sum of all positive divisors of 
$N$. In particular, these bounds hold if 
$N$ is an odd perfect number.
The main result in the earlier paper (by the first author) is improved as follows. The number of odd multiperfect numbers with at most 
$r$ distinct prime factors is bounded by 
${4}^{{r}^{2} } / {2}^{r+ 2} (r- 1)!$.
A natural number 
$n$ is called 
$k$-perfect if 
$\sigma (n)= kn$. In this paper, we show that for any integers 
$r\geq 2$ and 
$k\geq 2$, the number of odd 
$k$-perfect numbers 
$n$ with 
$\omega (n)\leq r$ is bounded by 
$\left({\lfloor {4}^{r} { \mathop{ \log } \nolimits }_{3} 2\rfloor + r\atop r} \right){ \mathop{ \sum } \nolimits }_{i= 1}^{r} \left({\lfloor kr/ 2\rfloor \atop i} \right)$, which is less than 
${4}^{{r}^{2} } $ when 
$r$ is large enough.
