prime omega function

The function $\backslash omega(n)$ is additive and $\backslash Omega(n)$ is completely additive. Little omega has the formula

$\backslash omega(n)=\backslash sum\_\{p\backslash mid\; n\}\; 1,$

where notation {{Math|p{{!}}n}} indicates that the sum is taken over all primes {{Mvar|p}} that divide {{Mvar|n}}, without multiplicity. For example, $\backslash omega(12)=\backslash omega(2^2\; 3)=2$.

Big omega has the formulas

$\backslash Omega(n)\; =\backslash sum\_\{p^\backslash alpha\backslash mid\; n\}\; 1\; =\backslash sum\_\{p^\backslash alpha\backslash parallel\; n\}\backslash alpha.$

The notation {{Math|p^α{{!}}n}} indicates that the sum is taken over all prime powers {{Math|p^α}} that divide {{Mvar|n}}, while {{Math|p^α{{!}}{{!}}n}} indicates that the sum is taken over all prime powers {{Math|p^α}} that divide {{Mvar|n}} and such that {{Math|n / p^α}} is coprime to {{Math|p^α}}. For example, $\backslash Omega(12)=\backslash Omega(2^2\; 3^1)=3$.

The omegas are related by the inequalities {{Math|ω(n) ≤ Ω(n)}} and {{Math|2^ω(n) ≤ d(n) ≤ 2^Ω(n)}}, where {{Math|d(n)}} is the divisor-counting function.This inequality is given in Section 22.13 of Hardy and Wright. If {{Math|1=Ω(n) = ω(n)}}, then {{Mvar|n}} is squarefree and related to the Möbius function by

:$\backslash mu(n)\; =\; (-1)^\{\backslash omega(n)\}\; =\; (-1)^\{\backslash Omega(n)\}.$

If $\backslash omega(n)\; =\; 1$ then $n$ is a prime power, and if $\backslash Omega(n)=1$ then $n$ is prime.

An asymptotic series for the average order of $\backslash omega(n)$ is S. R. Finch, Two asymptotic series, Mathematical Constants II, Cambridge Univ. Press, pp. 21-32, [https://web.archive.org/web/20160419150604/http://www.people.fas.harvard.edu/~sfinch/csolve/asym.pdf]

:$\backslash frac\{1\}\{n\}\; \backslash sum\backslash limits\_\{k\; =\; 1\}^n\; \backslash omega(k)\; \backslash sim\; \backslash log\backslash log\; n\; +\; B\_1\; +\; \backslash sum\_\{k\; \backslash geq\; 1\}\; \backslash left(\backslash sum\_\{j=0\}^\{k-1\}\; \backslash frac\{\backslash gamma\_j\}\{j!\}\; -\; 1\backslash right)\; \backslash frac\{(k-1)!\}\{(\backslash log\; n)^k\},$

where $B\_1\; \backslash approx\; 0.26149721$ is the Mertens constant and $\backslash gamma\_j$ are the Stieltjes constants.

The function $\backslash omega(n)$ is related to divisor sums over the Möbius function and the divisor function, including:Each of these started from the second identity in the list are cited individually on the pages Dirichlet convolutions of arithmetic functions, Menon's identity, and other formulas for Euler's totient function. The first identity is a combination of two known divisor sums cited in Section 27.6 of the [https://dlmf.nist.gov/27.6 NIST Handbook of Mathematical Functions].

:$\backslash sum\_\{d\backslash mid\; n\}\; |\backslash mu(d)|\; =\; 2^\{\backslash omega(n)\}$ is the number of unitary divisors. {{Oeis|A034444}}

:$\backslash sum\_\{d\backslash mid\; n\}\; |\backslash mu(d)|\; k^\{\backslash omega(d)\}\; =\; (k+1)^\{\backslash omega(n)\}$

:$\backslash sum\_\{r\backslash mid\; n\}\; 2^\{\backslash omega(r)\}\; =\; d(n^2)$

:$\backslash sum\_\{r\backslash mid\; n\}\; 2^\{\backslash omega(r)\}\; d\backslash left(\backslash frac\{n\}\{r\}\backslash right)\; =\; d^2(n)$

:$\backslash sum\_\{d\backslash mid\; n\}\; (-1)^\{\backslash omega(d)\}\; =\; \backslash prod\backslash limits\_\{p^\{\backslash alpha\}||n\}\; (1-\backslash alpha)$

:$$

\sum_{\stackrel{1\le k\le m}{(k,m)=1}} \gcd(k^2-1,m_1)\gcd(k^2-1,m_2)

=\varphi(n)\sum_{\stackrel{d_1\mid m_1} {d_2\mid m_2}} \varphi(\gcd(d_1, d_2)) 2^{\omega(\operatorname{lcm}(d_1, d_2))},\ m_1, m_2 \text{ odd}, m = \operatorname{lcm}(m_1, m_2)

:$\backslash sum\_\backslash stackrel\{1\backslash le\; k\backslash le\; n\}\{\backslash operatorname\{gcd\}(k,m)=1\}\; \backslash !\backslash !\backslash !\backslash !\; 1\; =\; n\; \backslash frac\; \{\backslash varphi(m)\}\{m\}\; +\; O\; \backslash left\; (\; 2^\{\backslash omega(m)\}\; \backslash right\; )$

The characteristic function of the primes can be expressed by a convolution with the Möbius function:This is suggested as an exercise in Apostol's book. Namely, we write $f\; =\; \backslash mu\; \backslash ast\; \backslash omega$ where $f(n)\; =\; \backslash sum\_\{d|n\}\; \backslash mu(n/d)\; \backslash sum\_\{r|d\}\; \backslash left(\backslash pi(r)\; -\; \backslash pi(r-1)\backslash right)$. We can form the Dirichlet series over $f$ as $D\_f(s)\; :=\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{f(n)\}\{n^s\}\; =\; P(s),$ where $P(s)$ is the prime zeta function. Then it becomes obvious to see that $f(n)\; =\; \backslash pi(n)\; -\; \backslash pi(n-1)\; =\; \backslash chi\_\{\backslash mathbb\{P\}\}(n)$ is the indicator function of the primes.

:$\backslash chi\_\{\backslash mathbb\{P\}\}(n)$

= (\mu \ast \omega)(n) = \sum_{d|n} \omega(d) \mu(n/d).

A partition-related exact identity for $\backslash omega(n)$ is given by This identity is proved in the article by Schmidt cited on this page below.

:$\backslash omega(n)\; =\; \backslash log\_2\backslash left[\backslash sum\_\{k=1\}^n\; \backslash sum\_\{j=1\}^k\; \backslash left(\backslash sum\_\{d\backslash mid\; k\}\; \backslash sum\_\{i=1\}^d\; p(d-ji)\; \backslash right)\; s\_\{n,k\}\; \backslash cdot\; |\backslash mu(j)|\backslash right],$

where $p(n)$ is the partition function, $\backslash mu(n)$ is the Möbius function, and the triangular sequence $s\_\{n,k\}$ is expanded by

:$s\_\{n,k\}\; =\; [q^n]\; (q;\; q)\_\backslash infty\; \backslash frac\{q^k\}\{1-q^k\}\; =\; s\_o(n,\; k)\; -\; s\_e(n,\; k),$

in terms of the infinite q-Pochhammer symbol and the restricted partition functions $s\_\{o/e\}(n,\; k)$ which respectively denote the number of $k$'s in all partitions of $n$ into an odd (even) number of distinct parts.This triangular sequence also shows up prominently in the Lambert series factorization theorems proved by Merca and Schmidt (2017–2018)

A continuation of $\backslash omega(n)$ has been found, though it is not analytic everywhere.{{Cite journal |last1=Hoelscher |first1=Zachary |last2=Palsson |first2=Eyvindur |date=2020-12-05 |title=Counting Restricted Partitions of Integers Into Fractions: Symmetry and Modes of the Generating Function and a Connection to ω(t) |url=https://journals.calstate.edu/pump/article/view/2428 |journal=The PUMP Journal of Undergraduate Research |language=en |volume=3 |pages=277–307 |doi=10.46787/pump.v3i0.2428 |arxiv=2011.14502 |issn=2576-3725}} Note that the normalized $\backslash operatorname\{sinc\}$ function $\backslash operatorname\{sinc\}(x)\; =\; \backslash frac\{\backslash sin(\backslash pi\; x)\}\{\backslash pi\; x\}$ is used.

:$\backslash omega(z)\; =\; \backslash log\_2\backslash left(\backslash sum\_\{n=1\}^\{\backslash lceil\; Re(z)\; \backslash rceil\}\; \backslash operatorname\{sinc\}\; \backslash left(\backslash prod\_\{m=1\}^\{\backslash lceil\; Re(z)\; \backslash rceil+1\}\; \backslash left(\; n^2+n-mz\; \backslash right)\; \backslash right)\; \backslash right)$

This is closely related to the following partition identity. Consider partitions of the form

:$a=\; \backslash frac\{2\}\{c\}\; +\; \backslash frac\{4\}\{c\}\; +\; \backslash ldots\; +\; \backslash frac\{2(b-1)\}\{c\}\; +\; \backslash frac\{2b\}\{c\}$

where $a$, $b$, and $c$ are positive integers, and $a\; >\; b\; >\; c$. The number of partitions is then given by $2^\{\backslash omega(a)\}\; -\; 2$. {{Cite journal |last1=Hoelscher |first1=Zachary |last2=Palsson |first2=Eyvindur |date=2020-12-05 |title=Counting Restricted Partitions of Integers Into Fractions: Symmetry and Modes of the Generating Function and a Connection to ω(t) |url=https://journals.calstate.edu/pump/article/view/2428 |journal=The PUMP Journal of Undergraduate Research |language=en |volume=3 |pages=277–307 |doi=10.46787/pump.v3i0.2428 |arxiv=2011.14502 |issn=2576-3725}}

An average order of both $\backslash omega(n)$ and $\backslash Omega(n)$ is $\backslash log\backslash log\; n$. When $n$ is prime a lower bound on the value of the function is $\backslash omega(n)\; =\; 1$. Similarly, if $n$ is primorial then the function is as large as

$\backslash omega(n)\; \backslash sim\; \backslash frac\{\backslash log\; n\}\{\backslash log\backslash log\; n\}$

on average order. When $n$ is a power of 2, then $\backslash Omega(n)\; =\; \backslash log\_2(n).$For references to each of these average order estimates see equations (3) and (18) of the MathWorld reference and Section 22.10-22.11 of Hardy and Wright.

Asymptotics for the summatory functions over $\backslash omega(n)$, $\backslash Omega(n)$, and powers of $\backslash omega(n)$ are respectivelySee Sections 22.10 and 22.11 for reference and explicit derivations of these asymptotic estimates.Actually, the proof of the last result given in Hardy and Wright actually suggests a more general procedure for extracting asymptotic estimates of the moments $\backslash sum\_\{n\; \backslash leq\; x\}\; \backslash omega(n)^k$ for any $k\; \backslash geq\; 2$ by considering the summatory functions of the factorial moments of the form $\backslash sum\_\{n\; \backslash leq\; x\}\; \backslash frac\{\backslash left[\backslash omega(n)\backslash right]!\}\{\backslash left[\backslash omega(n)-m\backslash right]!\}$ for more general cases of $m\; \backslash geq\; 2$.

:$\backslash begin\{align\}$

\sum_{n \leq x} \omega(n) & = x \log\log x + B_1 x + o(x) \\

\sum_{n \leq x} \Omega(n) & = x \log\log x + B_2 x + o(x) \\

\sum_{n \leq x} \omega(n)^2 & = x (\log\log x)^2 + O(x \log\log x) \\

\sum_{n \leq x} \omega(n)^k & = x (\log\log x)^k + O(x (\log\log x)^{k-1}), k \in \mathbb{Z}^{+},

\end{align}

where $B\_1\; \backslash approx\; 0.2614972128$ is the Mertens constant and the constant $B\_2$ is defined by

:$B\_2\; =\; B\_1\; +\; \backslash sum\_\{p\backslash text\{\; prime\}\}\; \backslash frac\{1\}\{p(p-1)\}\; \backslash approx\; 1.0345061758.$

The sum of number of unitary divisors is

$\backslash sum\_\{n\; \backslash le\; x\}\; 2^\{\backslash omega(n)\}\; =(x\; \backslash log\; x)/\backslash zeta(2)\; +\; O(x)${{Cite journal |last=Cohen |first=Eckford |date=1960 |title=The Number of Unitary Divisors of an Integer |url=https://www.jstor.org/stable/2309455 |journal=The American Mathematical Monthly |volume=67 |issue=9 |pages=879–880 |doi=10.2307/2309455 |jstor=2309455 |issn=0002-9890}} {{OEIS|A064608}}

Other sums relating the two variants of the prime omega functions include Hardy and Wright Chapter 22.11.

:$\backslash sum\_\{n\; \backslash leq\; x\}\; \backslash left\backslash \{\backslash Omega(n)\; -\; \backslash omega(n)\backslash right\backslash \}\; =\; O(x),$

and

:$\backslash \#\backslash left\backslash \{n\; \backslash leq\; x\; :\; \backslash Omega(n)\; -\; \backslash omega(n)\; >\; \backslash sqrt\{\backslash log\backslash log\; x\}\backslash right\backslash \}\; =\; O\backslash left(\backslash frac\{x\}\{(\backslash log\backslash log\; x)^\{1/2\}\}\backslash right).$

In this example we suggest a variant of the summatory functions $S\_\{\backslash omega\}(x)\; :=\; \backslash sum\_\{n\; \backslash leq\; x\}\; \backslash omega(n)$ estimated in the above results for sufficiently large $x$. We then prove an asymptotic formula for the growth of this modified summatory function derived from the asymptotic estimate of $S\_\{\backslash omega\}(x)$ provided in the formulas in the main subsection of this article above.N.b., this sum is suggested by work contained in an unpublished manuscript by the contributor to this page related to the growth of the Mertens function. Hence it is not just a vacuous and/or trivial estimate obtained for the purpose of exposition here.

To be completely precise, let the odd-indexed summatory function be defined as

:$S\_\{\backslash operatorname\{odd\}\}(x)\; :=\; \backslash sum\_\{n\; \backslash leq\; x\}\; \backslash omega(n)\; [n\backslash text\{\; odd\}],$

where $[\backslash cdot]$ denotes Iverson bracket. Then we have that

:$S\_\{\backslash operatorname\{odd\}\}(x)\; =\; \backslash frac\{x\}\{2\}\; \backslash log\backslash log\; x\; +\; \backslash frac\{(2B\_1-1)x\}\{4\}\; +\; \backslash left\backslash \{\backslash frac\{x\}\{4\}\backslash right\backslash \}\; -\; \backslash left[x\; \backslash equiv\; 2,3\; \backslash bmod\{4\}\backslash right]\; +\; O\backslash left(\backslash frac\{x\}\{\backslash log\; x\}\backslash right).$

The proof of this result follows by first observing that

:$$

\omega(2n) = \begin{cases}

\omega(n) + 1, & \text{if } n \text{ is odd; } \\

\omega(n), & \text{if } n \text{ is even,}

\end{cases}

and then applying the asymptotic result from Hardy and Wright for the summatory function over $\backslash omega(n)$, denoted by $S\_\{\backslash omega\}(x)\; :=\; \backslash sum\_\{n\; \backslash leq\; x\}\; \backslash omega(n)$, in the following form:

:$\backslash begin\{align\}$

S_\omega(x) & = S_{\operatorname{odd}}(x) + \sum_{n \leq \left\lfloor\frac{x}{2}\right\rfloor} \omega(2n) \\

& = S_{\operatorname{odd}}(x) + \sum_{n \leq \left\lfloor\frac{x}{4}\right\rfloor} \left(\omega(4n) + \omega(4n+2)\right) \\

& = S_{\operatorname{odd}}(x) + \sum_{n \leq \left\lfloor\frac{x}{4}\right\rfloor} \left(\omega(2n) + \omega(2n+1) + 1\right) \\

& = S_{\operatorname{odd}}(x) + S_{\omega}\left(\left\lfloor\frac{x}{2}\right\rfloor\right) + \left\lfloor\frac{x}{4}\right\rfloor.

\end{align}

The computations expanded in Chapter 22.11 of Hardy and Wright provide asymptotic estimates for the summatory function

:$\backslash omega(n)\; \backslash left\backslash \{\backslash omega(n)-1\backslash right\backslash \},$

by estimating the product of these two component omega functions as

:$\backslash omega(n)\; \backslash left\backslash \{\backslash omega(n)-1\backslash right\backslash \}\; =\; \backslash sum\_\{\backslash stackrel\{pq\backslash mid\; n\}\; \{\backslash stackrel\{p\; \backslash neq\; q\}\{p,q\backslash text\{\; prime\}\}\}\}\; 1\; =$

\sum_{\stackrel{pq\mid n}{p,q\text{ prime}}} 1 - \sum_{\stackrel{p^2\mid n}{p\text{ prime}}} 1.

We can similarly calculate asymptotic formulas more generally for the related summatory functions over so-termed factorial moments of the function $\backslash omega(n)$.

A known Dirichlet series involving $\backslash omega(n)$ and the Riemann zeta function is given by This identity is found in Section 27.4 of the [https://dlmf.nist.gov/27.4 NIST Handbook of Mathematical Functions].

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{2^\{\backslash omega(n)\}\}\{n^s\}\; =\; \backslash frac\{\backslash zeta^2(s)\}\{\backslash zeta(2s)\},\backslash \; \backslash Re(s)\; >\; 1.$

We can also see that

:

:

The function $\backslash Omega(n)$ is completely additive, where $\backslash omega(n)$ is strongly additive (additive). Now we can prove a short lemma in the following form which implies exact formulas for the expansions of the Dirichlet series over both $\backslash omega(n)$ and $\backslash Omega(n)$:

Lemma. Suppose that $f$ is a strongly additive arithmetic function defined such that its values at prime powers is given by $f(p^\{\backslash alpha\})\; :=\; f\_0(p,\; \backslash alpha)$, i.e., $f(p\_1^\{\backslash alpha\_1\}\; \backslash cdots\; p\_k^\{\backslash alpha\_k\})\; =\; f\_0(p\_1,\; \backslash alpha\_1)\; +\; \backslash cdots\; +\; f\_0(p\_k,\; \backslash alpha\_k)$ for distinct primes $p\_i$ and exponents $\backslash alpha\_i\; \backslash geq\; 1$. The Dirichlet series of $f$ is expanded by

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{f(n)\}\{n^s\}\; =\; \backslash zeta(s)\; \backslash times\; \backslash sum\_\{p\backslash mathrm\{\backslash \; prime\}\}\; (1-p^\{-s\})\; \backslash cdot\; \backslash sum\_\{n\; \backslash geq\; 1\}\; f\_0(p,\; n)\; p^\{-ns\},$

\Re(s) > \min(1, \sigma_f).

Proof. We can see that

:$\backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash frac\{u^\{f(n)\}\}\{n^s\}\; =\; \backslash prod\_\{p\backslash mathrm\{\backslash \; prime\}\}\; \backslash left(1+\backslash sum\_\{n\; \backslash geq\; 1\}\; u^\{f\_0(p,\; n)\}\; p^\{-ns\}\backslash right).$

This implies that

:$\backslash begin\{align\}$

\sum_{n \geq 1} \frac{f(n)}{n^s} & =

\frac{d}{du}\left[\prod_{p\mathrm{\ prime}} \left(1+\sum_{n \geq 1} u^{f_0(p, n)} p^{-ns}\right)\right] \Biggr|_{u=1}

=

\prod_{p} \left(1 + \sum_{n \geq 1} p^{-ns}\right) \times \sum_{p} \frac{\sum_{n \geq 1} f_0(p, n) p^{-ns}}{

1 + \sum_{n \geq 1} p^{-ns}} \\

& = \zeta(s) \times \sum_{p\mathrm{\ prime}} (1-p^{-s}) \cdot \sum_{n \geq 1} f_0(p, n) p^{-ns},

\end{align}

wherever the corresponding series and products are convergent. In the last equation, we have used the Euler product representation of the Riemann zeta function.

The lemma implies that for $\backslash Re(s)\; >\; 1$,

:$\backslash begin\{align\}$

D_{\omega}(s) & := \sum_{n \geq 1} \frac{\omega(n)}{n^s} = \zeta(s) P(s) \\

& \ = \zeta(s) \times \sum_{n \geq 1} \frac{\mu(n)}{n} \log \zeta(ns) \\

D_{\Omega}(s) & := \sum_{n \geq 1} \frac{\Omega(n)}{n^s} = \zeta(s) \times \sum_{n \geq 1} P(ns) \\

& \ = \zeta(s) \times \sum_{n \geq 1} \frac{\phi(n)}{n} \log\zeta(ns) \\

D_h(s) & := \sum_{n \geq 1} \frac{h(n)}{n^s} = \zeta(s) \log \zeta(s) \\

& \ = \zeta(s) \times \sum_{n \geq 1} \frac{\varepsilon(n)}{n} \log \zeta(ns),

\end{align}

where $P(s)$ is the prime zeta function, $h(n)\; =\; \backslash sum\_\{p^k|n\}\{\backslash frac\{1\}\{k\}\}\; =\; \backslash sum\_\{p^k||n\}\{H\_\{k\}\}$ where $H\_\{k\}$ is the $k$-th harmonic number and $\backslash varepsilon$ is the identity for the Dirichlet convolution, $\backslash varepsilon\; (n)\; =\; \backslash lfloor\backslash frac\{1\}\{n\}\backslash rfloor$.

The distribution of the distinct integer values of the differences $\backslash Omega(n)\; -\; \backslash omega(n)$ is regular in comparison with the semi-random properties of the component functions. For $k\; \backslash geq\; 0$, define

:$N\_k(x)\; :=\; \backslash \#(\backslash \{n\; \backslash in\; \backslash mathbb\{Z\}^\{+\}:\; \backslash Omega(n)\; -\; \backslash omega(n)\; =\; k\backslash \}\; \backslash cap\; [1,\; x]).$

These cardinalities have a corresponding sequence of limiting densities $d\_k$ such that for $x\; \backslash geq\; 2$

:$N\_k(x)\; =\; d\_k\; \backslash cdot\; x\; +\; O\backslash left(\backslash left(\backslash frac\{3\}\{4\}\backslash right)^k\; \backslash sqrt\{x\}\; (\backslash log\; x)^\{\backslash frac\{4\}\{3\}\}\backslash right).$

These densities are generated by the prime products

:$\backslash sum\_\{k\; \backslash geq\; 0\}\; d\_k\; \backslash cdot\; z^k\; =\; \backslash prod\_p\; \backslash left(1\; -\; \backslash frac\{1\}\{p\}\backslash right)\; \backslash left(1\; +\; \backslash frac\{1\}\{p-z\}\backslash right).$

With the absolute constant $\backslash hat\{c\}\; :=\; \backslash frac\{1\}\{4\}\; \backslash times\; \backslash prod\_\{p\; >\; 2\}\; \backslash left(1\; -\; \backslash frac\{1\}\{(p-1)^2\}\backslash right)^\{-1\}$,

the densities $d\_k$ satisfy

:$d\_k\; =\; \backslash hat\{c\}\; \backslash cdot\; 2^\{-k\}\; +\; O(5^\{-k\}).$

Compare to the definition of the prime products defined in the last section of {{cite journal|last1=Rényi|first1=A.|last2=Turán|first2=P.|title=On a theorem of Erdös-Kac|journal=Acta Arithmetica|volume=4|number=1|year=1958|pages=71–84|doi=10.4064/aa-4-1-71-84|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa417.pdf}} in relation to the Erdős–Kac theorem.

• Additive function
• Arithmetic function
• Erdős–Kac theorem
• Omega function (disambiguation)
• Prime number
• Square-free integer

{{Reflist|30em}}

• {{cite book|last1=G. H. Hardy and E. M. Wright|title=An Introduction to the Theory of Numbers|date=2006|publisher=Oxford University Press|edition=6th}}
• {{cite book|last1=H. L. Montgomery and R. C. Vaughan|title=Multiplicative number theory I. Classical theory|date=2007|publisher= Cambridge University Press|edition=1st}}
• {{cite arXiv|last1=Schmidt|first1=Maxie|title=Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions|year=2017|class=math.NT|eprint=1712.00608}}
• {{cite web|last1=Weisstein|first1=Eric|title=Distinct Prime Factors|url=http://mathworld.wolfram.com/DistinctPrimeFactors.html|website=MathWorld|access-date=22 April 2018}}

• [http://oeis.org/wiki/Omega(n),_number_of_distinct_primes_dividing_n OEIS Wiki for related sequence numbers and tables]
• [https://oeis.org/wiki/Prime_factors OEIS Wiki on Prime Factors]

Category:Number theory

Category:Prime numbers

Category:Additive functions

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