Os números suavemente ondulantes generalizados

In this article, for a fixed base d >= 2, we present and analyze some properties associated with the class of smoothly undulating numbers of the form [ab_n]_d, which we will call NSOG. We provide Binet's Formula for every NSOG. In particular, we study the divisibility or multiplicity relationship between two NSOG numbers. In NSOGs, those formed only by 1 and 0, in any base d >= 2, are highlighted, and we denote them by [10_n]_d. Regarding these numbers [10_n]_d, we show that none of them is prime in base 10. Furthermore, we present an algorithm for calculating the GCD between two numbers [10_n]_d, with n odd. Additionally, we show that the difference between two NSOG numbers is a perfect square. Finally, we present the connection between NSOGs and monodigit, repunit, and triangular numbers.