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A set of arithmetical sequences $$ a_1\, (\bmod{ \,\, m_1}) \quad, \quad a_2\, (\bmod{\,\, m_2}) \quad, \quad \dots \quad , \quad a_k \, (\bmod{\,\,m_k})\quad \quad , $$ with $$ m_1 \leq m_2 \leq \dots \leq m_k \quad \quad , $$ iscalled a {\it disjoint covering system} (alias {\it exact covering system}) ifevery positive integer belongs to {\bf exactly} one of the sequences. Mirski,Newman, Davenport and Rado famously proved that the moduli can't all bedistinct. In fact the two largest moduli must be equal, i.e. $m_{k-1}=m_k$ Thisraises the natural question:"How close can you get to getting distinctmoduli?", in other words, can you find all such systems where all the moduliare distinct except the largest, that is repeated $r$ times, for any, specificgiven $r$?  It turns out (conjecturally, but almost certainly) that excluding the trivialcase where the smallest modulus is 2, for any number of repeats $r$, there areonly finitely many such systems. Marc Berger, Alexander Felzenbaum and AviezriFraenkel found them all for $r$ up to $9$, and Mekmamu Zeleke and Jamie Simpsonextended the list for systems up to $12$ repeats. In the present article wecontinue the list up to $r=32$. All our systems are correct, but we did notbother to formally prove completeness, but we know for sure that the lists arecomplete if the largest modulus is $\leq 600$, and we are pretty sure that theyare complete.

Searching for Disjoint Covering Systems with Precisely One Repeated Modulus