What's the largest number of prime divisors that a number $n$ could have?

A natural number $n$ has 20 divisors. What's the largest number of different prime divisors that a number $n$ could have?

What I have is $$(\alpha_1+1)(\alpha_2+1)(\alpha_3+1)\cdots(\alpha_k+1)=20$$ where $$n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$$ But what's next?..

• You want to maximise $k$, so which factorisation of $20$ maximises the number of terms? – B. Mehta Jun 11 '17 at 13:59
• $\omega (n) < \frac{\ln n}{\ln \ln n} + O\left(\frac{\ln}{(\ln \ln n )^2}\right)$ and $\Omega(n) \le \log_2 n$ where $\omega(n)$ is the number of distinct prime divisors of $n$ and $\Omega(n)$ is the number of divisors of $n$ with multiplicity. – Dando18 Jun 11 '17 at 14:10
• In the text body you can italicize a piece of text by wrapping it with asterisks. Double asterisks give you bold. Alas, that doesn't seem to work in the title, but I think this looks better than using math italics that you had with TeX/MathJax. – Jyrki Lahtonen Jun 11 '17 at 16:48

For your case, the answer is $3$.

In general, if a number has $k$ divisors, what's the maximum number of prime divisors that it could have?

Every integer can be written as the product of different primes raised to some power greater than or equal to $1$, for example $x = {p_1}^{n_1} {p_2}^{n_2} \ldots {p_m}^{n_m}$, with all $n_j \geq 1$. The number of divisors is $k = (n_1 + 1)(n_2 + 1)\ldots(n_m + 1)$. The number of prime divisors is $m$.

$m$ is largest if the $n_j$ are as small as possible: Take $k$, factor it into prime factors, and count the prime factors according to their multiplicity. For example, if a number $x$ has $k = 20 = 2 \cdot 2 \cdot 5$ divisors, it can have at most $3$ prime factors since $2 \cdot 2 \cdot 5$ is the product of three primes. Numbers with $20$ divisors and $3$ prime divisors are the numbers of the form $pqr^4$ where $p$, $q$ and $r$ are distinct primes.

the number of factors in the expression $(\alpha_1+1)(\alpha_2+2)\dots(\alpha_k+1)=20$ is at most the number of prime factors of $20$ with multiplicity, which is $3$.

An example of such a factorization is $2\times2\times 5$ which yields the number $n=2\times 3\times 5^4$ for example.

• Sorry, I forgot to say, that these prime numbers should be different. – Karagum Jun 11 '17 at 14:05
• yeah, the maximum is $3$, for example $2,3$ and $5$. I just meant that the number of factors in the expression for $20$ is bounded by the number of priome factors of $20$ with multiplicity. – Jorge Fernández Hidalgo Jun 11 '17 at 14:09