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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 OEIS A001414]). ::''a''<sub>0</sub>(4) = 4
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