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

Abstract

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