Editing Practical number (section)

== The number of prime factors, the number of divisors, and the sum of divisors ==

The [[Erdős–Kac theorem]] implies that for a large random integer <math> n </math>, the number of prime factors of <math> n </math> (counted with or without multiplicity) follows an approximate [[normal distribution]] with mean <math> \log\log n </math> and variance <math> \log\log n </math>. The corresponding result for practical numbers<ref>{{harvtxt|Tenenbaum|Weingartner|2024}}</ref> implies that for a large random practical number <math> n </math>, the number of prime factors is approximately normal with mean <math> C \log\log n </math> and variance <math> V \log\log n </math>, where <math> C = 1/(1-e^{-\gamma}) = 2.280\ldots </math> and <math>V=0.414\ldots </math>. That is, most large integers <math> n </math> have about <math> \log\log n </math> prime factors, while most large practical numbers <math> n </math> have about <math> C \log\log n \approx 2.28 \log\log n </math> prime factors. 

As a consequence, most large integers <math> n </math>  have <math> 2^{(1+o(1))\log\log n} = (\log n)^{0.693\ldots} </math> divisors, while most large practical numbers <math> n </math> have <math> 2^{(C+o(1))\log\log n} = (\log n)^{1.580\ldots}</math>  divisors. In both cases, the average number of divisors is much larger than the typical number of divisors: for integers <math> n \le x </math>, the average number of divisors is about <math> \log x </math>, while for practical numbers <math> n \le x </math>, it is about <math> (\log x)^{1.713\ldots} </math>.<ref>{{harvtxt|Weingartner|2023}}</ref>

The average value of the sum-of-divisors function <math> \sigma(n) </math>, for integers <math> n \le x </math>, as well as for practical numbers <math> n \le x </math>, has order of magnitude <math> x </math>.<ref>Corollary 5 of {{harvtxt|Pomerance|Weingartner|2021}}</ref>