Editing Harmonic divisor number (section)

==Factorization of the harmonic mean==
The harmonic mean {{math|''H''(''n'')}} of the divisors of any number {{mvar|n}} can be expressed as the formula
<math display="block">H(n) = \frac{n\sigma_0(n)}{\sigma_1(n)}</math>
where {{math|''&sigma;''<sub>''i''&thinsp;</sub>(''n'')}} is the [[divisor function|sum of {{mvar|i}}th powers of the divisors]] of {{mvar|n}}: {{math|''&sigma;''<sub>0</sub>}} is the number of divisors, and {{math|''&sigma;''<sub>1</sub>}} is the sum of divisors {{harv|Cohen|1997}}.
All of the terms in this formula are [[multiplicative function|multiplicative]] but not [[Completely multiplicative function|completely multiplicative]].
Therefore, the harmonic mean {{math|''H''(''n'')}} is also multiplicative.
This means that, for any positive integer {{mvar|n}}, the harmonic mean {{math|''H''(''n'')}} can be expressed as the product of the harmonic means of the [[prime power]]s in the [[Integer factorization|factorization]] of {{mvar|n}}.

For instance, we have
<math display="block">H(4) = \frac{3}{1+\frac{1}{2}+\frac{1}{4}}=\frac{12}7,</math>
<math display="block">H(5) = \frac{2}{1+\frac{1}{5}} = \frac53,</math>
<math display="block">H(7) = \frac{2}{1+\frac{1}{7}} = \frac74,</math>
and
<math display="block">H(140) = H(4 \cdot 5 \cdot 7) = H(4)\cdot H(5)\cdot H(7) = \frac{12}{7}\cdot \frac{5}{3}\cdot \frac{7}{4} = 5.</math>