Dyck Numbers, I. Successor Function

Abstract: Dyck paths are among the most heavily studied Catalan families. In the paper we are dealing with the minimal numbering of Dyck paths, with the resulting numbers, the terms of the OEIS sequence A036991, which we have called Dyck numbers. We consider the successor function on the Dyck numbers, closed formulas are obtained. In particular, the formula for the successor of the Mersenne number is calculated (since OEIS A000225 is a subset of A036991). We have obtained the corresponding algorithm of the Dyck successor function, DS-function, and a Python program has been compiled. At the end of the paper, the size of the A036991 ranges is associated with the terms of OEIS A001405. The corresponding hypothesis is formulated.