Editing Additive function (section)

== Examples ==

Examples of arithmetic functions which are completely additive are:

* The restriction of the [[Logarithm|logarithmic function]] to <math>\N.</math>
* The '''multiplicity''' of a [[Prime number|prime]] factor ''p'' in ''n'', that is the largest exponent ''m'' for which ''p<sup>m</sup>'' [[Divisor|divides]] ''n''.
* {{anchor|Integer logarithm}} ''a''<sub>0</sub>(''n'') – the sum of primes dividing ''n'' counting multiplicity, sometimes called sopfr(''n''), the potency of ''n'' or the '''integer logarithm''' of ''n'' {{OEIS|A001414}}. For example:

::''a''<sub>0</sub>(4) = 2 + 2 = 4
::''a''<sub>0</sub>(20) =  ''a''<sub>0</sub>(2<sup>2</sup> · 5) = 2 + 2 + 5 = 9
::''a''<sub>0</sub>(27) = 3 + 3 + 3 = 9
::''a''<sub>0</sub>(144) = ''a''<sub>0</sub>(2<sup>4</sup> · 3<sup>2</sup>) = ''a''<sub>0</sub>(2<sup>4</sup>) + ''a''<sub>0</sub>(3<sup>2</sup>) = 8 + 6 = 14
::''a''<sub>0</sub>(2000) = ''a''<sub>0</sub>(2<sup>4</sup> · 5<sup>3</sup>) = ''a''<sub>0</sub>(2<sup>4</sup>) + ''a''<sub>0</sub>(5<sup>3</sup>) = 8 + 15 = 23
::''a''<sub>0</sub>(2003) = 2003
::''a''<sub>0</sub>(54,032,858,972,279) = 1240658
::''a''<sub>0</sub>(54,032,858,972,302) = 1780417
::''a''<sub>0</sub>(20,802,650,704,327,415) = 1240681

* The function Ω(''n''), defined as the total number of [[Prime factor#Omega functions|prime factors]] of ''n'', counting multiple factors multiple times, sometimes called the "Big Omega function" {{OEIS|A001222}}. For example;

::Ω(1) = 0, since 1 has no prime factors
::Ω(4) = 2
::Ω(16) = Ω(2·2·2·2) = 4
::Ω(20) = Ω(2·2·5) = 3
::Ω(27) = Ω(3·3·3) = 3
::Ω(144) = Ω(2<sup>4</sup> · 3<sup>2</sup>) = Ω(2<sup>4</sup>) + Ω(3<sup>2</sup>) = 4 + 2 = 6
::Ω(2000) = Ω(2<sup>4</sup> · 5<sup>3</sup>) = Ω(2<sup>4</sup>) + Ω(5<sup>3</sup>) = 4 + 3 = 7
::Ω(2001) = 3
::Ω(2002) = 4
::Ω(2003) = 1
::Ω(54,032,858,972,279) = Ω(11 ⋅ 1993<sup>2</sup> ⋅ 1236661) = 4
::Ω(54,032,858,972,302) = Ω(2 ⋅ 7<sup>2</sup> ⋅ 149 ⋅ 2081 ⋅ 1778171) = 6 
::Ω(20,802,650,704,327,415) = Ω(5 ⋅ 7 ⋅ 11<sup>2</sup> ⋅ 1993<sup>2</sup> ⋅ 1236661) = 7.

Examples of arithmetic functions which are additive but not completely additive are:

* ω(''n''), defined as the total number of distinct [[Prime factor#Omega functions|prime factors]] of ''n'' {{OEIS|A001221}}. For example:

::ω(4) = 1
::ω(16) = ω(2<sup>4</sup>) = 1
::ω(20) = ω(2<sup>2</sup> · 5) = 2
::ω(27) = ω(3<sup>3</sup>) = 1
::ω(144) = ω(2<sup>4</sup> · 3<sup>2</sup>) = ω(2<sup>4</sup>) + ω(3<sup>2</sup>) = 1 + 1 = 2
::ω(2000) = ω(2<sup>4</sup> · 5<sup>3</sup>) = ω(2<sup>4</sup>) + ω(5<sup>3</sup>) = 1 + 1 = 2
::ω(2001) = 3
::ω(2002) = 4
::ω(2003) = 1
::ω(54,032,858,972,279) = 3
::ω(54,032,858,972,302) = 5
::ω(20,802,650,704,327,415) = 5

* ''a''<sub>1</sub>(''n'') – the sum of the distinct primes dividing ''n'', sometimes called sopf(''n'') {{OEIS|A008472}}. For example:

::''a''<sub>1</sub>(1) = 0
::''a''<sub>1</sub>(4) = 2
::''a''<sub>1</sub>(20) = 2 + 5 = 7
::''a''<sub>1</sub>(27) = 3
::''a''<sub>1</sub>(144) = ''a''<sub>1</sub>(2<sup>4</sup> · 3<sup>2</sup>) = ''a''<sub>1</sub>(2<sup>4</sup>) + ''a''<sub>1</sub>(3<sup>2</sup>) = 2 + 3 = 5
::''a''<sub>1</sub>(2000) = ''a''<sub>1</sub>(2<sup>4</sup> · 5<sup>3</sup>) = ''a''<sub>1</sub>(2<sup>4</sup>) + ''a''<sub>1</sub>(5<sup>3</sup>) = 2 + 5 = 7
::''a''<sub>1</sub>(2001) = 55
::''a''<sub>1</sub>(2002) = 33
::''a''<sub>1</sub>(2003) = 2003
::''a''<sub>1</sub>(54,032,858,972,279) = 1238665
::''a''<sub>1</sub>(54,032,858,972,302) = 1780410
::''a''<sub>1</sub>(20,802,650,704,327,415) = 1238677