On RATS sequences in general bases (Q2926276)

\textit{J. H. Conway}'s [``Play it again and again\dots, Quantum 63, 30--31 November/December (1990)] RATS sequences in base \(b\) are generated from a positive integer by repeating the following process: reverse the order of the digits in base \(b\), add the result and the original number, then sort the sum's digits in increasing order from left to right.NEWLINENEWLINEThe author shows that for \(m>1\) and \(b=3\cdot 2^m-2\), there are infinitely many \(p\) such that there are periodic elements with period \(p\) in base \(b\). In fact, all or all even sufficiently large integers \(p\) have this property if \(m\) is even or odd, respectively.NEWLINENEWLINEThe sequence is said to be quasiperiodic if there is one digit (``the growing digit'') such that every \(q\)th iteration, the same digits occur with the same frequencies except for the growing digit, whose frequency increases by one. The least such integer \(q>0\) is called the quasiperiod.NEWLINENEWLINEIf \(q>2\) and \(b\equiv 1\mod{(2^q-1)^2}\) with \(b>1\), the author shows that there is a RATS sequence in base \(b\) with quasiperiod \(q\). It is also shown that there are no quasiperiodic RATS sequences with period \(1\). Furthermore, if \(b=2^t+1\) with \(t\geq 0\), there are no quasiperiodic RATS sequences in base \(b\).