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Omega(n), number of distinct primes dividing n
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(Redirected from Distinct primes version of Liouville's function)
n |
-
n = ω (n)∏ i = 1
ω (n) |
n |
n |
ω (44100) = ω (2 2 3 2 5 2 7 2 ) = 4 |
n |
For any positive value
k |
gcd (n, n + 1) = 1 |
gcd (n, n − 1) = 1 |
k |
A007018
a (0) = 1; a (n) = a (n − 1) (a (n − 1) + 1), n ≥ 1. |
- {1, 2, 6, 42, 1806, 3263442, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806, ...}
a (0) = 3; a (n) = a (n − 1) (a (n − 1) − 1), n ≥ 1. |
- {3, 6, 30, 870, 756030, 571580604870, 326704387862983487112030, 106735757048926752040856495274871386126283608870, ...}
Contents
Properties
ω (n) |
-
ω (m n) = ω (m) + ω (n), m ≥ 1, n ≥ 1, (m, n) = 1,
(m, n) |
m |
n |
Dirichlet generating function
The Dirichlet generating function of2 ω (n), n ≥ 1, |
-
D{2 ω (n)}(s) := ∞∑ n = 1
=2 ω (n) n s
, s > 1,ζ 2 (s) ζ (2 s)
ζ (s) |
Related arithmetic functions
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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 | |||||
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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 | |||||
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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 ofω (n) |
λ (n) := λ Ω (n) := ( − 1) Ω (n) |
Ω (n) |
-
λω (n) := (−1) ω (n)
ω (n) |
ω (n) |
Excess of n
A046660Ω (n) − ω (n), n ≥ 1, |
- {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 complementq̅ (n) := 1 − q (n) |
q (n) |
q̅ (n) := χnonsquarefree(n) = sgn [Ω (n) − ω (n)], n ≥ 1, |
sgn (n) |
Characteristic function of squarefree numbers
The quadratfrei functionq (n) := 1 − q̅ (n) := χsquarefree(n) = 1 − sgn [Ω (n) − ω (n)], n ≥ 1, |
sgn (n) |
Sequences
A001221 Number of prime factors of n (without multiplicity) (number of distinct prime factors of n):ω (n), n ≥ 1. |
- {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, ...}
|
- {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, ...}
λω (n) := ( − 1) ω (n), n ≥ 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, –1, ...}
Lω (n) :=
|
- {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, ...}
Ω (n), n ≥ 1. |
- {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, ...}
q̅ (n) := 1 − q (n) := χnonsquarefree(n) = sgn [Ω (n) − ω (n)], n ≥ 1. |
- {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, ...}
q (n) := 1 − q̅ (n) := χsquarefree(n) = 1 − sgn [Ω (n) − ω (n)], n ≥ 1. |
- {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 or prime factors of n (without multiplicity)
- {{Distinct prime factors}} or {{dpf}} arithmetic function template
- Number of distinct prime factors of n or number of prime factors of n (without multiplicity) (
)ω (n) - {{Number of distinct prime factors}} or {{little omega}} arithmetic function template
- Sum of distinct prime factors of n (
orsodpf (n)
)sopf (n) - {{Sum of distinct prime factors}} or {{sopf}} (sum of prime factors) arithmetic function template
- Product of distinct prime factors of n (squarefree kernel of n, radical of n) (
)rad (n) - {{Product of distinct prime factors}} or {{rad}} arithmetic function template
- Prime factors of n (with multiplicity)
- {{Prime factors (with multiplicity)}} or {{mpf}} arithmetic function template
- Number of prime factors of n (with multiplicity) (
)Ω (n) - {{Number of prime factors (with multiplicity)}} or {{big Omega}} arithmetic function template
- Sum of prime factors of n (with multiplicity) (
, sum of prime factors with repetition)sopfr (n) - {{Sum of prime factors (with multiplicity)}} or {{sopfr}} (sum of prime factors with repetition) arithmetic function template
- Product of prime factors of n (with multiplicity) (
, positive integers)n