Families of rational numbers with predictable Engel product expansions

In 1987 Knopfmacher and Knopfmacher published new infinite product expansions for real numbers 0 1. They called these Engel product expansions. At that time they had difficulty finding rational 0 < A < 1 for which the Engel product expansion is predictable. Later, in 1993, Arnold Knopfmacher presented many such families of rationals. In this paper we add to Arnold Knopfmacher's list of such families