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In [[number theory]], an '''additive function''' is an [[arithmetic function]] ''f''(''n'') of the positive [[integer]] ''n'' such that whenever ''a'' and ''b'' are [[coprime]] we have:
:''f''(''ab'') = ''f''(''a'') + ''f''(''b'').
An additive function ''f''(''n'') is said to be '''completely additive''' if ''f''(''ab'') = ''f''(''a'') + ''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime.
Every completely additive function is additive, but not vice versa.
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Arithmetic functions which are completely additive are:
* The restriction of the [[logarithm|logarithmic function]] to '''N'''.
* The function Ω(''n''), defined for every ''n'' ≥ 2 as the total number of [[prime number|prime]] factors of ''n'', counting multiple factors multiple times. We also agree to be Ω(1) = 0. Some values:
::Ω(4) = 2
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An example of an arithmetic function which is additive but not completely additive is
: ω(''n'') = ∑<sub>''p''|''n''</sub> 1(''n''),
for every positive integer ''n'', where sum runs over all different [[prime number|primes]] that devide ''n'' and 1(''n'') is a constant function, defined by 1(''n'') = 1. The ω function gives us the total number of ''different'' [[prime number|prime]] factors of arbitrary positive integer ''n''. Some values (compare with Ω(''n'')):
::ω(4) = 1
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