Intereting Posts

Deriving equations of motion in spherical coordinates
Why does this Fourier series have a finite number of terms?
Proof that 1-1 analytic functions have nonzero derivative
Subgroups of finite index have finitely many conjugates
Conditions for intersection of parabolas?
Complex Taylor and Laurent expansions
$ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
Does dividing by zero ever make sense?
Sum of the series: $\sum_{n=0}^\infty\frac{1+n}{3^n}$
Compendium(s) of Elementary Mathematical Truths
Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$
What is the difference between writing f and f(x)?
How can this integral expression for the difference between two $\zeta(s)$s be explained?
Inequality for small values of $t$
Proof of Drinker paradox

Specifically, I am thinking of a cuboid with a given volume ($28\,000$) that has sides of integer length. For example, $20 \cdot 20 \cdot 70 = 28\,000$, but so do $10 \cdot 40 \cdot 70$ and $1 \cdot 1 \cdot 28\,000$. I am interested in finding **how many possible integer combinations of side lengths there are that produce this volume**.

Its prime factorisation is $2^5 \cdot 5^3 \cdot 7$, so I think that the answer may have something to do with permutations of those.

The **order** of the three groups **does matter** because there is a distinction between it being height, width or length.

- How many fixed points in a permutation
- How many ways can we let people into a movie theater if they only have half-dollars and dollars?
- How many triangles are there?
- Probability for the length of the longest run in $n$ Bernoulli trials
- Number of solutions of $x_1+x_2+\dots+x_k=n$ with $x_i\le r$
- No closed form for the partial sum of ${n\choose k}$ for $k \le K$?

- A question about numbers from Euclid's proof of infinitude of primes
- Number of Derangements of the word BOTTLE
- Counting subsets containing three consecutive elements (previously Summation over large values of nCr)
- Explanation of the Fibonacci sequence appearing in the result of 1 divided by 89?
- Coloring 5 Largest Numbers in Each Row and Column Yields at Least 25 Double-Colored Numbers
- Choosing half of prefix and suffix
- Proving $\sum_{k=1}^nk^3 = \left(\sum_{k=1}^n k\right)^2$ using complete induction
- Connection Between Automorphism Groups of a Graph and its Line Graph
- $4$-digit positive integers that does not contain the digits $3$ and $4$ plus other properties
- Number of partitions of $n$ with $k$ parts equals the number of partitions of $n + \binom k {2}$

Suppose that $n=p_{1}^{k_1}\cdot p_{2}^{k_2}…p_{m}^{k_m}$. Then what we must do is decide is how to split up each prime power as the sum of $3$ non-negative integers. The way to do that is stars-and-bars and the formula becomes $\binom{k_{i}+2}{2}$. (This gives us the number of ways of splitting up the factor $p_{i}^{k_{i}}$.) All together, this gives us $\prod_{i=1}^{m}\binom{k_{i}+2}{2}$ ways to write $n$ as a product of exactly $3$ of its factors.

To be more explicit in why this works, think of writing your $3$ bins, $\ell, w, h$. $\ell$ will contain $p_{i}^{x_{1}}$, $w$ will contain $p_{i}^{x_2}$ and $h$ will contain $p_{i}^{x_{3}}$. In order for $\ell w h=n$ you need that $p_{i}^{x_1}p_{i}^{x_2}p_{i}^{x_3}=p_{i}^{k_{i}}$ so it must be that $x_1+x_2+x_3=k_i$ and each $x_{1},x_{2},x_{3}\geq 0$. Then, do this for each prime.

Using the formalism of the following MSE link and the Polya Enumeration Theorem it follows that for $n$ having factorization $$n = \prod_{p|n} p^v$$

applying PET we have almost by inspection that the desired count of factorizations into $k$ factors is given by the term

$$H(n, k) = \left[\prod_p X_p^v\right]

Z(E_k)\left(\prod_p \frac{1}{1-X_p}\right)$$

where the square bracket denotes coefficient extraction

of formal power series and $Z(E_k) = a_1^k$ is the cycle index of the identity group containing the identity permutation. This becomes

$$\left[\prod_p X_p^v\right]

\left(\prod_p \frac{1}{1-X_p}\right)^k.$$

Doing the coefficient extraction we obtain

$$\prod_p {v+k-1\choose k-1}$$

as observed by the first responder.

- Assigning values to divergent series
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- Rationalizing expressions
- Uniformly continuous function acts almost like an Lipschitz function?
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- What is a number?
- Prove by induction $\sum\limits_{k=m}^{\ n}{n\choose k}{k\choose m}={n\choose m}2^{n-m}$
- How to compute the elements of the field $\mathbb{R} / \langle x^2+1\rangle $ ?
- Can we always choose the generators of an ideal of a Noetherian ring to be homogeneous?
- To compute $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2|^2 dz$ where $\mathcal{C}$ is the unit circle in $\mathbb{C}$
- Finding the point on an ellipse most distant from a given line
- Order topology on the set $X = \{ 1,2 \} \times \mathbb{Z}_{+}$
- How to get ${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$
- Property of critical point when the Hessian is degenerate
- Probability of an odd number in 10/20 lotto