the golomb-dickman constant also arises when we consider the average length of the largest cycle of any function from a finite set to itself. if "x" is a finite set, if we repeatedly apply a function "f": "x" → "x" to any element "x" of this set, it eventually enters a cycle, meaning that for some "k" we have formula_10 for sufficiently large "n"; the smallest "k" with this property is the length of the cycle. let "b" be the average, taken over all functions from a set of size "n" to itself, of the length of the largest cycle. then purdom and williams proved that