Additive function: Difference between revisions

Revision as of 21:32, 19 September 2002

Revision as of 23:01, 19 September 2002

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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''), defined as the total number of ''different'' [[prime number|prime]] factors of ''n''. Some values (compare with Ω(''n'')):

: ω(''n'') = ∑_''p''|''n'' 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