Omega Function Calculator ω(n)

Omega Function Calculator
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 2ω(n) however is multiplicative:

2ω(mn) = 2ω(m)+ω(n) = (2ω(m))(2ω(n)).

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

Identities

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

The number of square-free divisors of n is 2ω(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 2ω(n). And the divisors of P are precisely the square-free divisors of n.

An infinite sum identity involving ω(n) is

    Σ 2ω(n)/ns = 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/ns = ζ(s)


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

G(n) = Σ 2ω(d)

where d ranges over the divisors of n. If n = p1a1⋅p2a2⋅...⋅pkak, where the pi's are the prime factors and the ai's are their respective powers, then

G(n) = Π (2ai + 1)

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

Some Special Values of ω(n)

For any power of 10, ω(10x) = 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.

nNumber of 1'sω(n)
110
1121
11132
111142
1111152
11111165
111111172
1111111184
11111111193
1111111111104
11111111111112
111111111111127
1111111111111133
11111111111111144
111111111111111156
1111111111111111166
11111111111111111172
111111111111111111188
1111111111111111111191
11111111111111111111207
111111111111111111111217
1111111111111111111111226
11111111111111111111111231
1111111111111111111111112410
1111111111111111111111111255
11111111111111111111111111266
111111111111111111111111111275
1111111111111111111111111111288
11111111111111111111111111111295
1111111111111111111111111111113013
1111111111111111111111111111111313
111111111111111111111111111111113211
111111111111111111111111111111111336
1111111111111111111111111111111111346
11111111111111111111111111111111111357

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