EA: "S = 22,55,628,5527,54140,52707,539525,5296736. Can you do better (more terms with another start)?" I think one can do better even with the same start. I wrote a procedure that will generate a list of all possible follow-up numbers for a given list of starting numbers. So: {22} -> {55} -> {226, 427, 628, 829} -> {5143, 5315, 5527, 5779} -> {51799, 54140} -> {52707} -> {137417, 237944, 338471, 438998, 539525, 640052, 740579, 841106, 941633} -> {1513071, 2140811, 2521482, 3529893, 4153612, 4538304, 5107073, 5296736, 5546715, 6166413, 6555126, 7563537, 8179214, 8571948, 9580359} -> {5630370, 11860703, 21906086, 28813922, 31951469, 41996852, 49005529, 52042235, 54260828, 57286149, 62087618, 69197136, 72133001, 82178384, 89388743, 92223767} -> and so on. You can see that extending S is only a matter of picking the right choice or choices of the possibles. There is a chance of course that at some point there will be no follow-up numbers but for this particular example I have not found that to be the case. Counting the number of integers in the lists, I get: 1, 1, 4, 4, 2, 1, 9, 15, 16, 31, 42, 62, 97, 136, 229, 411, 731, 1244, 2151, 3624, 6569, 11184, 19487, 33107, 57879, 99275, 173604, 299558, 510698, 886078, 1521519, ... {20} is a smaller start that has the appearance of possibly going on forever. My integer counts for it are: 1, 1, 4, 4, 4, 2, 1, 9, 5, 5, 8, 17, 32, 43, 70, 125, 208, 369, 580, 983, 1729, 2878, 4858, 8204, 14000, 25261, 46854, 84816, 143941, 245238, 423392, 731067, 1271021, 2186308, 3764677, 6463595, ...