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AN ANALYSIS OF n-EIVEN NUMBERS

For a positive integer a and w>2, define sn(a) to be the sum of the digits in the base n expansion of a. If sn is applied recursively, it clearly stabilizes at some value. Let S„(a) = s£(a) for all sufficiently large k. A Niven number [3] is a positive integer a that is divisible by $m(a). We define a riven number (short for recursive Niven number) to be a positive integer a that is divisible by Sl0(q). As in [2], these concepts are generalized to w-Miven numbers and w-riven numbers, using the functions sn and S„, respectively. In [1], Cooper and Kennedy proved that there does not exist a sequence of more than 20 consecutive Niven numbers and that this bound is optimal. Wilson [4] determined the digit sum of the smallest number initiating a maximal Niven number sequence. The author [2] proved that, for each n>2, there does not exist a sequence of more than 2n consecutive w-Niven numbers and Wilson [5] proved that this bound is optimal. This paper presents general properties of w-riven numbers and examines the maximal possible lengths of sequences of consecutive w-riven numbers. We begin with a basic lemma characterizing the value of Sn(a), which leads to many general facts about n-r'wm numbers. In Section 3 we determine the maximal lengths of sequences of consecutive «-riven numbers. We construct examples of sequences of maximal length for each n including ones that are provably as small as possible in terms of the values of the numbers in them.