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 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.
Outside number theory, the term additive is usually used for all functions with the property f(ab) = f(a) + f(b) for all arguments a and b. This article discusses number theoretic additive functions.
Arithmetic functions which are completely additive are:
a_{0}(4) = 4
a_{0}(27) = 9
a_{0}(144) = a_{0}(2^{4} · 3^{2}) = a_{0}(2^{4}) + a_{0}(3^{2}) = 8 + 6 = 14
a_{0}(2,000) = a_{0}(2^{4} · 5^{3}) = a_{0}(2^{4}) + a_{0}(5^{3}) = 8 + 15 = 23
a_{0}(2,001) = 55
a_{0}(2,002) = 33
a_{0}(2,003) = 2003
a_{0}(54,032,858,972,279) = 1240658
a_{0}(54,032,858,972,302) = 1780417
a_{0}(20,802,650,704,327,415) = 1240681
...
- a_{1}(n) - the sum of the distinct primes dividing n, sometimes called sopf(n). We have a_{1}(1) = 0, a_{1}(20) = 2 + 5 = 7. Some more values: (SIDN A008472)
a_{1}(4) = 2
a_{1}(27) = 3
a_{1}(144) = a_{1}(2^{4} · 3^{2}) = a_{1}(2^{4}) + a_{1}(3^{2}) = 2 + 3 = 5
a_{1}(2,000) = a_{1}(2^{4} · 5^{3}) = a_{1}(2^{4}) + a_{1}(5^{3}) = 2 + 5 = 7
a_{1}(2,001) = 55
a_{1}(2,002) = 33
a_{1}(2,003) = 2003
a_{1}(54,032,858,972,279) = 1238665
a_{1}(54,032,858,972,302) = 1780410
a_{1}(20,802,650,704,327,415) = 1238677
...
- The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times. This implies Ω(1) = 0 since 1 has no prime factors. Some more values: (SIDN A001222)
Ω(4) = 2
Ω(27) = 3
Ω(144) = Ω(2^{4} · 3^{2}) = Ω(2^{4}) + Ω(3^{2}) = 4 + 2 = 6
Ω(2,000) = Ω(2^{4} · 5^{3}) = Ω(2^{4}) + Ω(5^{3}) = 4 + 3 = 7
Ω(2,001) = 3
Ω(2,002) = 4
Ω(2,003) = 1
Ω(54,032,858,972,279) = 3
Ω(54,032,858,972,302) = 6
Ω(20,802,650,704,327,415) = 7
...
- An example of an arithmetic function which is additive but not completely additive is ω(n), defined as the total number of different prime factors of n. Some values (compare with Ω(n)) (SIDN A001221)
- ω(4) = 1
- ω(27) = 1
- ω(144) = ω(2^{4} · 3^{2}) = ω(2^{4}) + ω(3^{2}) = 1 + 1 = 2
- ω(2,000) = ω(2^{4} · 5^{3}) = ω(2^{4}) + ω(5^{3}) = 1 + 1 = 2
- ω(2,001) = 3
- ω(2,002) = 4
- ω(2,003) = 1
- ω(54,032,858,972,279) = 3
- ω(54,032,858,972,302) = 5
- ω(20,802,650,704,327,415) = 5
- ...
Sources:
- Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp 97 - 108) (MSC (2000) 11A25)