Omega Function Calculator ω(n)

In number theory, the number of distinct prime factors of an integer n is denoted by ω(n). For example, ω(6) = ω(12) = ω(18) = ω(24) = 2 since each of these numbers is the product of powers of 2 and 3.

Unlike other number theoretic functions, ω(n) is not multiplicative, however, it does satisfy the relation

ω(mn) = ω(m) + ω(n)

for relatively prime integers m and n. This is similar to the logarithm identity log(ab) = log(a) + log(b). The function 2^ω(n) however is multiplicative:

2^ω(mn) = 2^ω(m)+ω(n) = (2^ω(m))(2^ω(n)).

The omega function plays a role in several number theoretic identities.

Identities

For any positive integers n and x, ω(n^x) = ω(n).

The number of square-free divisors of n is 2^ω(n). To see why this is so, let P be the largest square-free divisor of n. P is simply the product of all the primes that divide n. Put another way, if you factor n into the products of its prime powers, P is the number you get when you replace every exponent with 1. Using the formula for σ₀(P), the number of divisors of P is 2^ω(n). And the divisors of P are precisely the square-free divisors of n.

An infinite sum identity involving ω(n) is

    Σ 2^ω(n)/n^s = ζ(s)²/ζ(2s),

where the n ranges from 1 to infinity and ζ(s) is the Riemann Zeta function given by

    Σ 1/n^s = ζ(s)


Let the multiplicative function G(n) be defined by expression

G(n) = Σ 2^ω(d)

where d ranges over the divisors of n. If n = p₁^a1⋅p₂^a2⋅...⋅p_k^ak, where the p_i's are the prime factors and the a_i's are their respective powers, then

G(n) = Π (2a_i + 1)

where i ranges from 1 to k. (Necessarily, k = ω(n)).

Some Special Values of ω(n)

For any power of 10, ω(10^x) = 2 since 10 = 2 times 5. For factorials, ω(n!) simply equals the number of primes less than or equal to n. Example: ω(20!) = 8 since there are exactly 8 primes that do not exceed 20: 2, 3, 5, 7, 11, 13, 17, and 19.

The table below gives ω(n) for n = 1, 11, 111, etc.

© Had2Know 2010