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Concept
Additive function
Summary
In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b:
An additive function f(n) is said to be completely additive if holds for all positive integers a and b, even when they are not coprime. 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.
Examples of arithmetic functions which are completely additive are:
The restriction of the logarithmic function to
The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n.
a0(n) – the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n . For example:
a0(4) = 2 + 2 = 4
a0(20) = a0(22 · 5) = 2 + 2 + 5 = 9
a0(27) = 3 + 3 + 3 = 9
a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14
a0(2000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
a0(2003) = 2003
a0(54,032,858,972,279) = 1240658
a0(54,032,858,972,302) = 1780417
a0(20,802,650,704,327,415) = 1240681
The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" . For example;
Ω(1) = 0, since 1 has no prime factors
Ω(4) = 2
Ω(16) = Ω(2·2·2·2) = 4
Ω(20) = Ω(2·2·5) = 3
Ω(27) = Ω(3·3·3) = 3
Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6
Ω(2000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7
Ω(2001) = 3
Ω(2002) = 4
Ω(2003) = 1
Ω(54,032,858,972,279) = Ω(11 ⋅ 19932 ⋅ 1236661) = 4 ;
Ω(54,032,858,972,302) = Ω(2 ⋅ 72 ⋅ 149 ⋅ 2081 ⋅ 1778171) = 6
Ω(20,802,650,704,327,415) = Ω(5 ⋅ 7 ⋅ 112 ⋅ 19932 ⋅ 1236661) = 7.
Examples of arithmetic functions which are additive but not completely additive are:
ω(n), defined as the total number of distinct prime factors of n .
Official source
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