Additive function
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, the function of the product is the sum of the functions:
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 co-prime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0.
Every completely additive function is additive, but not vice versa.
Example of arithmetic functions which are completely additive are:
Example of arithmetic functions which are additive but not completely additive are:
From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have:
One such example is g(n) = 2f(n).
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