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Jacobi symbol; quadratic nonresidue; clock sequence; primitive Šindel sequences; Chinese remainder theorem; Dirichlet's theorem
We found that there is a remarkable relationship between the triangular numbers $T_k$ and the astronomical clock (horologe) of Prague. We introduce Šindel sequences $\{a_i\}\subset \Bbb N$ of natural numbers as those periodic sequences with period $p$ that satisfy the following condition: for any $k\in\Bbb N$ there exists $n\in\Bbb N$ such that $T_k=a_1+\cdots+a_n$. We shall see that this condition guarantees a functioning of the bellworks, which is controlled by the horologe. We give a necessary and sufficient condition for a periodic sequence to be a Šindel sequence. We also present an algorithm which produces the so-called primitive Šindel sequence, which is uniquely determined for a given $s=a_1+\cdots+a_p$.
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