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Omega(n), number of distinct primes dividing n

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The canonical prime factorization of
n
being
n = \prod_{i=1}^{\omega(n)} {p_i}^{\alpha_i}, \,
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
.

Contents

Properties

ω (n)
is an additive arithmetic function, i.e.
\omega(mn) = \omega(m) + \omega(n),\ m \,\ge\, 1,\quad n \,\ge\, 1, (m,n) = 1, \,
where
(m, n)
is the greatest common divisor of
m
and
n
.

Dirichlet generating function

The Dirichlet generating function of
2ω (n), n ≥ 1,
is
D_{\{2^{\omega(n)}\}}(s) \equiv \sum_{n=1}^{\infty} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta^2(s)}{\zeta(2s)},\ 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 \scriptstyle \omega(n)\,, (Liouville's function being \scriptstyle \lambda(n) \,\equiv\, {\lambda}_{\Omega}(n) \,\equiv\, (-1)^{\Omega(n)}\, for \scriptstyle \Omega(n)\,, the total number of primes dividing n)

{\lambda}_{\omega}(n) \equiv (-1)^{\omega(n)}\,
is +1 when
ω (n)
is even and -1 when
ω (n)
is odd.

Excess of n

A046660 \scriptstyle \Omega(n) \,-\, \omega(n),\ n \,\ge\, 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 \scriptstyle \bar{q}(n) \,=\, 1 \,-\, q(n)\, of the quadratfrei function \scriptstyle q(n)\,, \scriptstyle \bar{q}(n) \,\equiv\, \chi_{\{nonsquarefree\}}(n) \,\equiv\, \sgn [\Omega(n) \,-\, \omega(n)],\ n \,\ge\, 1,\, is the characteristic function of nonsquarefree numbers, \scriptstyle \sgn(n)\, being the sign function.

Characteristic function of squarefree numbers

The quadratfrei function \scriptstyle q(n) \,\equiv\, 1 \,-\, \bar{q}(n) \,\equiv\, \chi_{\{squarefree\}}(n) \,\equiv\, 1 \,-\, \sgn [\Omega(n) \,-\, \omega(n)],\ n \,\ge\, 1\, is the characteristic function of squarefree numbers, \scriptstyle \sgn(n)\, being the sign function.

Sequences

A001221 \scriptstyle \omega(n),\ n \,\ge\, 1,\, 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 \scriptstyle \sum_{i=1}^{n} \omega(i),\ n \,\ge\, 1, \, 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?????? \scriptstyle {\lambda}_{\omega}(n) \,\equiv\, (-1)^{\omega(n)},\ n \,\ge\, 1, \, "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 \scriptstyle L_{\omega}(n) \,\equiv\, \sum_{i=1}^{n} {\lambda}_{\omega}(i) \,\equiv\, \sum_{i=1}^{n} (-1)^{\omega(i)},\ n \,\ge\, 1, \, "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 \scriptstyle \Omega(n),\ n \,\ge\, 1, \, 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 \scriptstyle \bar{q}(n) \,\equiv\, \chi_{\{nonsquarefree\}}(n) \,\equiv\, \sgn [\Omega(n) \,-\, \omega(n)],\ n \,\ge\, 1, \, (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 \scriptstyle q(n) \,\equiv\, 1 \,-\, \bar{q}(n) \,\equiv\, \chi_{\{squarefree\}}(n) \,\equiv\, 1 \,-\, \sgn [\Omega(n) \,-\, \omega(n)],\ n \,\ge\, 1, \, (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.




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