We study Hoeffding decomposable exchangeable sequences with values in a finite set D = {d1, . . . , dK}. We provide a new combinatorial characterization of Hoeffding decomposability and use this result to show that, for every K ≥ 3, there exists a class of neither Pólya nor i.i.d. D-valued exchangeable sequences that are Hoeffding decomposable.

In Sárközy (2001) the third author of this paper presented two conjectures on additive decomposability sequence ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) second (less demanding) conjecture. The goal is to extend sharpen their result in three directions by a different approach (based theory S-unit equations).