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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.

Examples

Arithmetic functions which are completely additive are:

  • The restriction of the logarithmic function to N, a0(n) - the sum of primes dividing n, sometimes called sopfr(n). We have a0(20) = a0(22 · 5) = 2 + 2+ 5 = 9. Some values: (SIDN A001414 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001414)).

a0(4) = 4
a0(27) = 9
a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14
a0(2,000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
a0(2,001) = 55
a0(2,002) = 33
a0(2,003) = 2003
a0(54,032,858,972,279) = 1240658
a0(54,032,858,972,302) = 1780417
a0(20,802,650,704,327,415) = 1240681
...

  • a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n). We have a1(1) = 0, a1(20) = 2 + 5 = 7. Some more values: (SIDN A008472 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472))

a1(4) = 2
a1(27) = 3
a1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5
a1(2,000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7
a1(2,001) = 55
a1(2,002) = 33
a1(2,003) = 2003
a1(54,032,858,972,279) = 1238665
a1(54,032,858,972,302) = 1780410
a1(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 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001222))

Ω(4) = 2
Ω(27) = 3
Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6
Ω(2,000) = Ω(24 · 53) = Ω(24) + Ω(53) = 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 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001221))

ω(4) = 1
ω(27) = 1
ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2
ω(2,000) = ω(24 · 53) = ω(24) + ω(53) = 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:

  1. Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp 97 - 108) (MSC (2000) 11A25)



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