Odd integers $N$ with five distinct prime factors for which $2-10^{-12}<\sigma (N)/N<2+10^{-12}$

Abstract:

We make a table of odd integers N with five distinct prime factors for which $2 - {10^{ - 12}} < \sigma (N)/N < 2 + {10^{ - 12}}$, and show that for such $N\;|\sigma (N)/N - 2| > {10^{ - 14}}$. Using this inequality, we prove that there are no odd perfect numbers, no quasiperfect numbers and no odd almost perfect numbers with five distinct prime factors. We also make a table of odd primitive abundant numbers N with five distinct prime factors for which $2 < \sigma (N)/N < 2 + 2/{10^{10}}$.