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Arithmetic functions which are completely additive are:
* The restriction of the [[logarithm|logarithmic function]] to '''N''', ''a''<sub>0</sub>(''n'') - the sum of primes dividing ''n'', sometimes called sopfr(''n''). We have ''a''<sub>0</sub>(20) = ''a''<sub>0</sub>(2<sup>2</sup> · 5) = 2 + 2+ 5 = 9. Some values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001414 SIDN A001414]).
* The function Ω(''n''), defined as the total number of [[prime number|prime]] factors of ''n'', counting multiple factors multiple times. This implies Ω(1) = 0 since 1 has no prime factors. Some values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001222 SIDN A001222])▼
::''a''<sub>0</sub>(4) = 4
::''a''<sub>0</sub>(27) = 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>(2,000) = ''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>(2,001) = 55
::''a''<sub>0</sub>(2,002) = 33
::''a''<sub>0</sub>(2,003) = 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
:: ...
* ''a''<sub>1</sub>(''n'') - the sum of the distinct primes dividing ''n'', sometimes called sopf(''n''). We have ''a''<sub>1</sub>(1) = 0, ''a''<sub>1</sub>(20) = 2 + 5 = 7. Some more values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472 SIDN A008472])
::''a''<sub>1</sub>(4) = 2
::''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>(2,000) = ''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>(2,001) = 55
::''a''<sub>1</sub>(2,002) = 33
::''a''<sub>1</sub>(2,003) = 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
:: ...
▲* The function Ω(''n''), defined as the total number of [[prime number|prime]] factors of ''n'', counting multiple factors multiple times. This implies Ω(1) = 0 since 1 has no prime factors. Some more values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001222 SIDN A001222])
::Ω(4) = 2
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:: ...
* An example of an arithmetic function which is additive but not completely additive is ω(''n''), defined as the total number of ''different'' [[prime number|prime]] factors of ''n''. Some values (compare with Ω(''n'')) ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001221 SIDN A001221])
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