Additive Function - Examples

Examples

Example of arithmetic functions which are completely additive are:

• The restriction of the logarithmic function to N.

• 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 (sequence A001414 in OEIS). 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(2,000) = a₀(24 · 53) = a₀(24) + a₀(53) = 8 + 15 = 23

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

• 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" (sequence A001222 in OEIS). For example;

Ω(1) = 0, since 1 has no prime factors

Ω(20) = Ω(2·2·5) = 3

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

Example of arithmetic functions which are additive but not completely additive are:

• ω(n), defined as the total number of different prime factors of n (sequence A001221 in OEIS). For example:

ω(4) = 1

ω(20) = ω(22·5) = 2

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

• a₁(n) - the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence A008472 in OEIS). For example:

a₁(1) = 0

a₁(4) = 2

a₁(20) = 2 + 5 = 7

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