List of long open, elementary problems which are computational in nature I would like to ask a question of a similar vein to this question.
Question: I'm asking for a list of long open problems which are computational in nature which a beginning graduate student can understand. One problem per answer, please.
Meaning of "beginning graduate student": anyone who can solve all the problems on a pure mathematics qualifying exam at a top 30 institution in the U.S.    
Meaning of "computational in nature": By this, I do not mean a computational task which can be executed by a computer, but rather a problem where one must compute some object (e.g. topological invariant, closed formula, etc.) associated to some mathematical object. Example: calculating the homotopy groups of a sphere.
Meaning of "not too famous": (Same as in this question.): Roughly, if there exists a whole monograph already dedicated to the problem (or narrow circle of problems), no need to mention it again here. I'm looking for problems that, with high probability, a mathematician working outside the particular area has never encountered. 
Meaning of "long open": (Same as in this question): The problem should occur in the literature or have a solid history as folklore. So I do not mean to call here for the invention of new problems or to collect everybody's laundry list of private-research-impeding unproved elementary technical lemmas. There should already exist at least of small community of mathematicians who will care if one of these problems gets solved.
 A: I think the determinant spectrum problem should be more well known.
I've written about this elsewhere on MathOverflow, e.g.
Determinants of binary matrices .  Briefly:
Fix n, look at the set of n by n binary (I like 0-1 matrices, others prefer 1,-1, but they are morally equivalent) matrices, and compute their determinants over the integers. What is the set of values thus obtained? This is open for n=11, 13, and larger. (Unfortunately, Will Orrick's website at Indiana.edu is down at present, so you have to find an archived copy. The index n shifts by one sometimes, so it may be reported as 12, and 14 or larger.)
There are related questions, one of which might be computationally resolved: find a brief description, uniform in the parameter n, which gives better than exponentially many matrices whose spectrum subset is large and contiguous.  I got exponentially many which hit all determinants in (-2F(n-1),2F(n-1)) using Fibonacci matrices; can someone do better?
Gerhard "Thanks Again To Roger House" Paseman, 2020.04.24.
A: Černý conjecture was stated in 1964 and it's not very famous (no monograph, but a special number of Journal of Automata, Languages and Combinatorics last year), but probably is not "computational in nature", strictly speaking. Anyway, there are many open problems related to such conjecture which are also less known or studied.
E.g., let $f_1, \ldots, f_m$ be functions from $\{1,\ldots, n+1\}$ to itself, and let $h$ a function obtaind by composing $f_1, \ldots, f_m$ as many times as you want, so $h$ is a word on the alphabet $\{f_1, \ldots, f_m\}$. If the image of $h$ has cardinality $1$, then the set $\{f_1, \ldots, f_m\}$ is $n$-compressible and $h$ is a $n$-collapsing word.
Sauer and Stone proved that there exist words like $h$ that are $n$-collapsing for every $n$-compressible set of $m$ functions from $\{1,\ldots, n+1\}$ to itself: such words are called $n$-synchronizing words.
Find $s(n,m)$, the lenght of the shortest $n$-synchronizing word over an alphabet with $m$ functions (letters).
This is obviously "computational in nature" as for fixed $n$ and $m$ there is only a finite number of $n$-compressible sets $\{f_1, \ldots, f_m\}$ and an upper bound for $s(n,m)$ it's known.
We know that $s(3,2)=33$ and $s(2,3)=22$, try to find other values. (Note that the problem can be stated in a more effective way using automata, see here for many other details and the upper bound for $s(n,m)$).
A: I think this one fits the profile, since it computational in nature, understandable by an undergrad student and still an open problem:

The envy-free cake-cutting: the problem of cutting a heterogeneous "cake" that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation.
How many queries are required for cutting this cake into $n$ slices?

Whether it is "not too famous" might be disputable. Please take it with a grain of salt (I have never heard of it until a while ago, but I am not a mathematician). According to wikipedia and this other question:

The continuous "moving knife" algorithms for envy free cake cutting into connected pieces is only mentioned for up to 4 players. The general case is still an open problem.

A: Problem: extend the table of known van der Waerden numbers from 7 to 8 entries.
Given $K\geq 2$ colors, the length $N=W(L,K)$ of the smallest set of colored integers $\{1,2,3,\ldots N\}$ with a monochromatic arithmetic progression of length $L\geq 3$ is only known in 7 cases.  
 
The seventh entry on the list was computed in 2012: $W(3,4)=293$, meaning 293 is the smallest integer $N$ such that whenever the set of integers $\{1,2,3,\ldots N\}$ is 3-colored, there exists a monochromatic arithmetic progression of length 4.
Adding one more entry to this table seems to meet the four criteria in the OP: a problem which is "understandable", "computational", "not too famous" (unlike the Ramsey numbers), "long open" (van der Waerden's paper, which started the search for $W(2,L)$, is from 1927).
A: The moving sofa problem is really not too famous and it seems to be open at least since 1966.
This problem is surely computational in nature since it asks for the value of the sofa constant, which, as it seems, is at present unknown.
A: Note sure whether it is too famous (it has a monograph).

Find the Moore graph of girth $5$ and degree $57$, if one exists.
That means, find a graph with diameter $=2$ (i.e., the distance between any two vertices is at most two), girth $=5$ (i.e., the shortest cycle has length five) and degree $=57$ (i.e., any vertex has exactly 57 neighbors).
All Moore graphs are known, except this one.
If it exists, it must have 3250 vertices, so still quite accessible.
A: Finding the set of forbidden minors for the class of toroidal graphs (finite graphs that can be embedded in the torus with no crossings).  By the Robertson–Seymour theorem, this set is finite, but it is only partially known, and the finiteness proof is ineffective.  A recent paper by Myrvold and Woodcock states, among other things, that the current list of known obstructions (over 17000 forbidden minors!) is unlikely to be complete.
A: There exists a (99,14,1,2)-strongly regular graph? That is a graph with 99 vertices, each vertex connected with 14 other vertices, each edge entering in a unique triangle, and such that for each non-connected pair of vertices $a$, $b$, there exist other two $c$ and $d$, and only those two, connected simultaneously with $a$ and $b$? 
All the restrictions studied do not rule out the existence, but nobody has been able to construct it.  E. Berlekamp, J. H. van Lint y J. J. Seidel have constructed a (243,22,1,2)-strongly regular graph. (A strong Regular Graph Derived from the Perfect Ternary Golay Code, in the book A Survey of Cominatorial Theory,  ed. by J. N. Srivastava, North Holland, 1973, p.~25–30.)
A: I believe (I'm not a professional mathematician) that the problem concerning aliquot sequences could fit your requirements. Wikipedia has the article Aliquot sequence and the online encyclopedia Wolfram MathWorld has the article (Catalan's Aliquot Sequence Conjecture and) Aliquot Sequence both provide remarkable  references.
In my view two important articles that I've known in the past are [1] and [2]. If I refer well, I've  known it in the past, the professor Juan Luis Varona (Universidad de La Rioja) has a page/website dedicated to this subject.
For example this conjecture was as an answer on this Mathoverflow from the post What are conjectures that are true for primes but then turned out to be false for some composite number?, question with identificator 117891 on MathOverflow (Jan 2 '13), where is added more information in a concise way.
References:
[1] Richard K. Guy and J. L. Selfridge, What Drives an Aliquot Sequence?, Mathematics of Computation, Vol. 29, No. 129, (January, 1975), pp. 101-107.
[2] P. Erdös, On Asymptotic Properties of Aliquot Sequences, Mathematics of Computation, Vol. 30, No. 135, (July, 1976), pp. 641-645.
A: Are there any algebraic integers of degree $d \geq 3$ with bounded partial quotients? 
It is a theorem of Dirichlet that for every irrational number $\alpha$, there exists infinitely many rational numbers $p/q$ with $\gcd(p,q) = 1$ and $q > 0$ such that 
$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^2}.$ 
This can be improved with the constant on the right hand side improved to $1/\sqrt{5}$, and then this is sharp (Hurwitz's theorem). The reason is that there are badly approximable numbers, whose partial quotients in their continued fraction expansions are bounded, for which it is possible to prove a lower bound of the form $|\alpha - p/q| \geq c(\alpha) q^{-2}$ for all rational numbers $p/q$ and the constant $c(\alpha)$ depending only on $\alpha$. Note that since quadratic irrationals have eventually periodic continued fraction expansion, all quadratic irrationals are badly approximable. 
It is a theorem of Roth, for which he was awarded a Fields Medal in 1958, that for any algebraic integer $\alpha$ having degree $d \geq 2$ and for any $\varepsilon > 0$, the number of reduced fractions $p/q$ such that
$\displaystyle \left \lvert \alpha - \frac{p}{q} \right \rvert < \frac{1}{q^{2 + \varepsilon}}$
is finite. In other words, all algebraic integers are almost badly approximable. 
The question is, are there any algebraic integers $\alpha$ having degree $d \geq 3$ which has bounded partial quotients, or equivalently, badly approximable? This question, shockingly, remains open even for degree 3. 
