Editing Multiply perfect number (section)

== Bounds ==

In [[Big O notation#Little-o notation|little-o notation]], the number of multiply perfect numbers less than ''x'' is <math>o(x^\varepsilon)</math> for all ε > 0.<ref name="HBI105">{{harvnb|Sándor|Mitrinović|Crstici|2006|p=105}}</ref>

The number of ''k''-perfect numbers ''n'' for ''n'' ≤ ''x'' is less than <math>cx^{c'\log\log\log x/\log\log x}</math>, where ''c'' and ''c''' are constants independent of ''k''.<ref name="HBI105" />

Under the assumption of the [[Riemann hypothesis]], the following [[inequality (mathematics)|inequality]] is true for all {{nowrap|''k''-perfect}} numbers ''n'', where ''k'' > 3
:<math>\log\log n > k\cdot e^{-\gamma}</math>
where <math>\gamma</math> is [[Euler–Mascheroni constant|Euler's gamma constant]]. This can be proven using [[Robin's theorem]].

The [[number of divisors]] τ(''n'') of a {{nowrap|''k''-perfect}} number ''n'' satisfies the inequality<ref>{{cite arXiv |last=Dagal |first=Keneth Adrian P. |eprint=1309.3527 |title=A Lower Bound for τ(n) for k-Multiperfect Number |class=math.NT |date=2013}}</ref>
:<math>\tau(n) > e^{k - \gamma}.</math>

The [[prime omega function|number of distinct prime factors]] ω(''n'') of ''n'' satisfies<ref name="HBI106">{{harvnb|Sándor|Mitrinović|Crstici|2006|p=106}}</ref>
:<math>\omega(n) \ge k^2-1.</math>

If the distinct prime factors of ''n'' are <math>p_1, p_2, \ldots, p_r</math>, then:<ref name="HBI106" />
:<math>r \left(\sqrt[r]{3/2} - 1\right) < \sum_{i=1}^{r} \frac{1}{p_i} < r \left(1 - \sqrt[r]{6/k^2}\right), ~~ \text{if }n\text{ is even}</math>
:<math>r \left(\sqrt[3r]{k^2} - 1\right) < \sum_{i=1}^{r} \frac{1}{p_i} < r \left(1 - \sqrt[r]{8/(k\pi^2)}\right), ~~ \text{if }n\text{ is odd}</math>