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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
::''a''<sub>0</sub>(4) = 4
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:: ...
* ''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
::''a''<sub>1</sub>(4) = 2
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* The function Ω(''n''), defined as the total number of [[prime number|prime]] factors of ''n'', counting multiple factors multiple times. It is often called "[[Big Omega function]]".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
::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
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