Michal K\v r\'{\i }\v zek, Alena \v Solcov\'a, Lawrence Somer
Construction of \v Sindel sequences

Comment.Math.Univ.Carolin. 48,3 (2007) 373-388.

Abstract:We found that there is a remarkable relationship between the triangular numbers $T_k$ and the astronomical clock (horologe) of Prague. We introduce \v Sindel 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 \v Sindel sequence. We also present an algorithm which produces the so-called primitive \v Sindel sequence, which is uniquely determined for a given $s=a_1+\cdots +a_p$.

Keywords: Jacobi symbol, quadratic nonresidue, clock sequence, primitive \v Sindel sequences, Chinese remainder theorem, Dirichlet's theorem
AMS Subject Classification: 11A07, 11A51, 01A40