On the combinatorial structure of 3N + 1 predecessor sets

AbstractThe 3n + 1 predecessor set P(a) of an integer a consists of all integers n whose iterates Ti(n) eventually hit a, where T is the 3n + 1 function defined by T(n) = n/2 if n is even, T(n) = (3n + 1)/2 if n is odd. This paper gives a representation of the sets P(a) by certain sets of finite integer sequences and studies their combinatorial structure. The notion of small sequences is introduced and explored in connection with the sets P(a). We also investigate an averaging and approximation heuristics based on small sequences