we know the first odd near-perfect number is $3^4*7^2*11^2*19^2$(of course the number must be square).and from one paper ,which is studying the odd perfect number and giving some estimate on it, we can also use the inequality to describe the odd near-perfect number,such as $n=\prod_{i=1}^{r}p_{i}^{n_{i}}$is an odd near-perfect number,then we have $n>\frac{1}{(2^\frac{1}{r}-1)^r},\sum_{i=1}^{r}\frac{1}{p_i}<1$.is there any good estimate about the odd near-perfect number?and is there any more odd near-perfect number?
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3$\begingroup$ === Definition? === $\endgroup$– Włodzimierz HolsztyńskiCommented Apr 23, 2016 at 4:34
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1$\begingroup$ @WłodzimierzHolsztyński --- see answer below; this is evidently the definition implied by the OP, but there are others who call a number $n$ "near-perfect" if the sum of proper divisors equals $n-1$. $\endgroup$– Carlo BeenakkerCommented Apr 23, 2016 at 9:27
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1$\begingroup$ @CarloBeenakker That other definition is often called almost perfect numbers. $\endgroup$– Jeppe Stig NielsenCommented Apr 23, 2016 at 10:45
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$\begingroup$ I am enthusiastic about almost nearly quasi sub-perfect pseudointegers. $\endgroup$– Włodzimierz HolsztyńskiCommented Apr 23, 2016 at 23:42
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Indeed, so far the only known odd near-perfect number (a number $n$ with redundant divisor $d$ such that the sum of the proper divisors of $n$ is $n+d$) is the number found by Donovan Johnson, $173369889=3^4 \cdot 7^2 \cdot 11^2 \cdot 19^2$, see On perfect and near-perfect numbers by Paul Pollack and Vladimir Shevelev.
According to OEIS, it remains the only odd near-perfect number up to $1.4\cdot 10^{19}$
answered Apr 23, 2016 at 6:16