[math-fun] a(n) and a(n+1) substrings of their product

Hello Math-Fun, Carole Dubois and myself have tried to extend S -- but could not find a new term < 9 999 999 999. S = 1, 10, 11, 100, 21, 1000, 31, 10000, 41, 100000, 51, 1000000, 61, 10000000, 71, 100000000, 81, 2817. S should be the lexicographically earliest seq of distinct positive terms such that a(n) and a(n+1) are substrings of their product. Example: 81 * 2817 = 228177. Does S end there? Would a different a(1) produce more terms? Is there another way to exploit this idea of two terms visible in their product? (respectively addition, division) Or is this old hat? Best, É.