Editing Arithmetic function (section)

== Ω(''n''), ''ω''(''n''), ''ν''<sub>''p''</sub>(''n'') – prime power decomposition ==
The [[fundamental theorem of arithmetic]] states that any positive integer ''n'' can be represented uniquely as a product of powers of primes: <math> n = p_1^{a_1}\cdots p_k^{a_k} </math> where  ''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>''k''</sub> are primes and the ''a<sub>j</sub>'' are positive integers. (1 is given by the empty product.)

It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define the [[p-adic valuation|''p''-adic valuation]] '''ν<sub>''p''</sub>(''n'')''' to be the exponent of the highest power of the prime ''p'' that divides ''n''. That is, if ''p'' is one of the ''p''<sub>''i''</sub> then ''ν''<sub>''p''</sub>(''n'') = ''a''<sub>''i''</sub>, otherwise it is zero.  Then
<math display="block">n = \prod_p p^{\nu_p(n)}.</math>

In terms of the above the [[prime omega function]]s ''ω'' and Ω are defined by
{{block indent | em = 1.5 | text = ''ω''(''n'') = ''k'',}}
{{block indent | em = 1.5 | text = Ω(''n'') = ''a''<sub>1</sub> + ''a''<sub>2</sub> + ... + ''a''<sub>''k''</sub>.}}

To avoid repetition, formulas for the functions listed in this article are, whenever possible, given in terms of ''n'' and the corresponding ''p''<sub>''i''</sub>, ''a''<sub>''i''</sub>, ''ω'', and Ω.