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We make a table of odd integers  N  with five distinct prime factors for which      2 −  10 −  12 &gt; σ  ( N ) / N &gt; 2 + 10 −  12 2 - {10^{ - 12}} &gt; \sigma (N)/N &gt; 2 + {10^{ - 12}}      , and show that for such      N | σ  ( N ) / N −  2 | &gt; 10 −  14 N\;|\sigma (N)/N - 2| &gt; {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 &gt; σ  ( N ) / N &gt; 2 + 2 / 10 10 2 &gt; \sigma (N)/N &gt; 2 + 2/{10^{10}}      .

Odd integers 𝑁 with five distinct prime factors for which 2-10⁻¹²<𝜎(𝑁)/𝑁<2+10⁻¹²