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Omega(n), number of distinct primes dividing n
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n 
ω (n) 
n 
n = 44100 = (3⋅7) 2 (2⋅5) 2 = 2 2 3 2 5 2 7 2 
ω (44100) = ω (2 2 3 2 5 2 7 2) = 4 
n 
2, 3, 5 
7 
Contents 
Properties
ω (n) 
(m, n) 
m 
n 
Dirichlet generating function
The Dirichlet generating function of2 ω (n), n ≥ 1, 
ζ (s) 
Related arithmetic functions
 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  
 0  1  1  1  1  2  1  1  1  2  1  2  1  2  2  1  1  2  1  2  2  2  1  2  1  2  1  2  1  3  1  1  2  2  2  2  1  2  2  2  
 0  1  2  3  4  6  7  8  9  11  12  14  15  17  19  20  21  23  24  26  28  30  31  33  34  36  37  39  40  43  44  45  47  49  51  53  54  56  58  60  
 1  –1  –1  –1  –1  1  –1  –1  –1  1  –1  1  –1  1  1  –1  –1  1  –1  1  1  1  –1  1  –1  1  –1  1  –1  –1  –1  –1  1  1  1  1  –1  1  1  1  
 1  0  –1  –2  –3  –2  –3  –4  –5  –4  –5  –4  –5  –4  –3  –4  –5  –4  –5  –4  –3  –2  –3  –2  –3  –2  –3  –2  –3  –4  –5  –6  –5  –4  –3  –2  –3  –2  –1  0 
"Distinct primes version of Liouville's function"
The "distinct primes version of Liouville's function", expressing the parity of , (Liouville's function being for , the total number of primes dividing n)
ω (n) 
ω (n) 
Excess of n
A046660 excess of n = number of prime factors of n (with multiplicity)  number of prime factors of n (without multiplicity).

{0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, ...}
Characteristic function of nonsquarefree numbers
The complement of the quadratfrei function , is the characteristic function of nonsquarefree numbers, being the sign function.
Characteristic function of squarefree numbers
The quadratfrei function is the characteristic function of squarefree numbers, being the sign function.
Sequences
A001221 number of prime factors of n (without multiplicity), number of distinct prime factors of n.

{0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, ...}
A013939 The partial sums summatory omega function.

{0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 34, 36, 37, 39, 40, 43, 44, 45, 47, 49, 51, 53, 54, 56, 58, 60, 61, 64, 65, 67, 69, 71, 72, 74, 75, ...}
A?????? "distinct primes version of Liouville's function."

{1, –1, –1, –1, –1, 1, –1, –1, –1, 1, –1, 1, –1, 1, 1, –1, –1, 1, –1, 1, 1, 1, –1, 1, –1, 1, –1, 1, –1, –1, –1, –1, 1, 1, 1, 1, –1, 1, 1, 1, –1, –1, –1, 1, 1, 1, –1, 1, –1, 1, 1, 1, –1, 1, 1, 1, 1, 1, –1, –1, ...}
A?????? The partial sums "summatory distinct primes version of Liouville's function."

{1, 0, –1, –2, –3, –2, –3, –4, –5, –4, –5, –4, –5, –4, –3, –4, –5, –4, –5, –4, –3, –2, –3, –2, –3, –2, –3, –2, –3, –4, –5, –6, –5, –4, –3, –2, –3, –2, –1, 0, –1, –2, –3, –2, –1, 0, –1, 0, –1, 0, 1, 2, 1, 2, 3, 4, ...}
A001222 number of prime factors of n (with multiplicity).

{0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, ...}
A107078 (0, or 1 if n has nonunitary prime divisors), nonquadratfrei function, characteristic function of nonsquarefree numbers.

{0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, ...}
A008966 (0, or 1 if n has unitary prime divisors only), quadratfrei function, characteristic function of squarefree numbers.

{1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...}
See also
 A056912 Odd squarefree numbers for which the number of prime divisors is odd.
 A056913 Odd squarefree numbers for which the number of prime divisors is even.
 Distinct prime factors of n
 Number of distinct prime factors of n (ω (n))
 Sum of distinct prime factors of n (sopf(n) or sodpf(n))
 Product of distinct prime factors of n (radical of n, rad(n)) (squarefree kernel of n)
 Prime factors of n (with multiplicity)
 Number of prime factors of n (with multiplicity) (Ω (n))
 Sum of prime factors of n (with multiplicity) (sopfr(n)) (integer log of n)
 Product of prime factors of n (with multiplicity) (
, positive integers)n
 {{Distinct prime factors}} or {{dpf}} (mathematical function template)
 {{Prime factors (with multiplicity)}} or {{mpf}} (mathematical function template)
 {{Number of distinct prime factors}} or {{little omega}} (mathematical function template)
 {{Number of prime factors (with multiplicity)}} or {{big Omega}} (mathematical function template)