Editing Practical number (section)

== Analogies with prime numbers ==

One reason for interest in practical numbers is that many of their properties are similar to properties of the [[prime numbers]]. 
Indeed, theorems analogous to [[Goldbach's conjecture]] and the [[twin prime conjecture]] are known for practical numbers: every positive even integer is the sum of two practical numbers, and there exist infinitely many triples of practical numbers <math>(x-2,x,x+2)</math>.<ref>{{harvtxt|Melfi|1996}}.</ref> [[Giuseppe Melfi|Melfi]] also showed<ref>{{harvtxt|Melfi|1995}}</ref> that there are infinitely many practical [[Fibonacci number]]s {{OEIS|id=A124105}}; and Sanna<ref>{{harvtxt|Sanna|2019}}</ref> proved that at least <math>C n / \log n</math> of the first <math>n</math> terms of every [[Lucas sequence]] are practical numbers, where <math>C > 0</math> is a constant and <math>n</math> is sufficiently large. The analogous questions of the existence of infinitely many [[Fibonacci prime]]s, or prime in a Lucas sequence, are open. {{harvtxt|Hausman|Shapiro|1984}} showed that there always exists a practical number in the interval <math>[x^2,(x+1)^2)]</math> for any positive real <math>x</math>, a result analogous to [[Legendre's conjecture]] for primes. Moreover, for all sufficiently large <math>x</math>, the interval <math>[x-x^{0.4872},x]</math> contains many practical numbers.<ref>{{harvtxt|Weingartner|2022}}.</ref>

Let <math>p(x)</math> count how many practical numbers are at {{nowrap|most <math>x</math>.}}
{{harvtxt|Margenstern|1991}} conjectured that <math>p(x)</math> is asymptotic to <math>cx/\log x</math> for some constant <math>c</math>, a formula which resembles the [[prime number theorem]], strengthening the earlier claim of {{harvtxt|Erdős|Loxton|1979}} that the practical numbers have density zero in the integers.
Improving on an estimate of {{harvtxt|Tenenbaum|1986}}, {{harvtxt|Saias|1997}} found that <math>p(x)</math> has order of magnitude <math>x/\log x</math>.
{{harvtxt|Weingartner|2015}} proved Margenstern's conjecture. We have<ref>{{harvtxt|Weingartner|2015}} and Remark 1 of {{harvtxt|Pomerance|Weingartner|2021}}</ref>
<math display=block>p(x) = \frac{c x}{\log x}\left(1 + O\!\left(\frac{1}{\log x}\right)\right),</math>
where <math>c=1.33607...</math><ref>{{harvtxt|Weingartner|2020}}.</ref> Thus the practical numbers are about 33.6% more numerous than the prime numbers. The exact value of the constant factor <math>c</math> is given by<ref>{{harvtxt|Weingartner|2019}}.</ref>
<math display=block> c= \frac{1}{1-e^{-\gamma}} \sum_{n \ \text{practical}} \frac{1}{n}  \Biggl( \sum_{p\le \sigma(n)+1}\frac{\log p}{p-1} - \log n\Biggr) \prod_{p\le \sigma(n)+1} \left(1-\frac{1}{p}\right),</math>
where <math>\gamma</math> is the [[Euler–Mascheroni constant]] and <math>p</math> runs over primes.

As with prime numbers in an arithmetic progression, given two natural numbers <math>a</math> and <math>q</math>, 
we have<ref>{{harvtxt|Weingartner|2021}}</ref>
<math display=block>
|\{ n \le x: n \text{ practical and } n\equiv a \bmod q \}|=\frac{c_{q,a} x}{\log x} +O_q\left(\frac{x}{(\log x)^2}\right).
</math>
The constant factor <math>c_{q,a}</math> is positive if, and only if, there is more than one practical number congruent to <math> a \bmod q </math>.
If <math>\gcd(q,a)=\gcd(q,b)</math>, then <math>c_{q,a}=c_{q,b}</math>. 
For example, about 38.26% of practical numbers have a last decimal digit of 0, while the last digits of 2, 4, 6, 8 each occur with the same relative frequency of 15.43%.