Omega(n), number of distinct primes dividing n

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The canonical prime factorization of

n

being

n  =    ω (n) ∏   i   = 1    pi αi ,

where the function

ω (n)

is the number of distinct prime factors of the positive integer

n

, each prime factor being counted only once. For example, for

n

= 44100 = (3 ⋅  7 ) 2 (2 ⋅  5) 2 = 2 2 3 2 5 2 7 2 we have

ω (44100) = ω (2 2 3 2 5 2 7 2 ) = 4

, as the four distinct primes factors of

n

are 2, 3, 5 and 7.

For any positive value

k

, since

gcd (n, n + 1) = 1

and

gcd (n, n  −  1) = 1

, the following sequences give constructive proofs that there exists integers with at least

k

distinct prime factors.

A007018

a (0) = 1; a (n) = a (n  −  1) (a (n  −  1) + 1), n   ≥   1.

{1, 2, 6, 42, 1806, 3263442, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806, ...}

A117805

a (0) = 3; a (n) = a (n  −  1) (a (n  −  1)  −  1), n   ≥   1.

{3, 6, 30, 870, 756030, 571580604870, 326704387862983487112030, 106735757048926752040856495274871386126283608870, ...}

Properties

ω (n)

is an additive arithmetic function, i.e.

ω (m n)  =  ω (m) + ω (n), m ≥ 1, n ≥ 1, (m, n) = 1,

where

(m, n)

is the greatest common divisor of

m

and

n

.

Dirichlet generating function

The Dirichlet generating function of

2 ω (n), n   ≥   1,

is

D{2 ω (n)}(s)  :=   ∞ ∑   n   = 1    2 ω (n) n  s  =  ζ  2 (s) ζ  (2 s) , s > 1,

where

ζ  (s)

is the Riemann zeta function (Hardy and Wright 1979, p. 255).

Related arithmetic functions

Related arithmetic functions

n	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
ω (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
∑ n i  = 1  ω (i)	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) ω (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
∑ n i  = 1  ( − 1) ω (i)	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

ω (n)

, (Liouville’s function being

λ (n)   :=   λ Ω (n)   :=   ( − 1) Ω (n)

for

Ω (n)

, the number of prime factors of n (with multiplicity))

λω (n)  :=  (−1) ω (n)

is +1 when

ω (n)

is even and  − 1 when

ω (n)

is odd.

Excess of n

A046660

Ω (n)  −  ω (n), n   ≥   1,

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

q̅  (n)   :=   1  −  q (n)

of the quadratfrei function

q (n)

,

q̅  (n)   :=   χnonsquarefree(n) = sgn [Ω (n)  −  ω (n)], n   ≥   1,

is the characteristic function of nonsquarefree numbers,

sgn (n)

being the sign function.

Characteristic function of squarefree numbers

The quadratfrei function

q (n)   :=   1  −  q̅  (n)   :=   χsquarefree(n) = 1  −  sgn [Ω (n)  −  ω (n)], n   ≥   1,

is the characteristic function of squarefree numbers,

sgn (n)

being the sign function.

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, ...}

A013939 Summatory ω function: partial sums

∑ n i   = 1 ω (i ), n   ≥   1.

{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”:

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, ...}

A?????? “Summatory distinct primes version of Liouville’s function”: partial sums

Lω (n)   :=   ∑ n i   = 1 λω (i ) = ∑ n i   = 1 ( − 1) ω (i ), n   ≥   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, –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):

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, ...}

A107078 Nonquadratfrei function (characteristic function of nonsquarefree numbers):

q̅  (n)   :=   1  −  q (n)   :=   χnonsquarefree(n) = sgn [Ω (n)  −  ω (n)], n   ≥   1.

(0, or 1 if n has nonunitary prime divisors.)

{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 Quadratfrei function (characteristic function of squarefree numbers):

q (n)   :=   1  −  q̅  (n)   :=   χsquarefree(n) = 1  −  sgn [Ω (n)  −  ω (n)], n   ≥   1.

(0, or 1 if n has unitary prime divisors only.)

{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  (

  sodpf (n)

  or

  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)  (

  sopfr (n)

  , sum of prime factors with repetition)
  • {{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)  (

  n

  , positive integers)