A better program filled in missing values of loop length, caught 2 errors in my manual work (for s,b = 5,4 and 7,3), and extended the results to s (number of digits) and b (radix) = 16. (All of these start with s-1 zeros followed by a one.) s=2 s=3 s=4 s=5 s=6 s=7 s=8 s=9 s=10 b=2 3 3 1 3 2 1 1 1 5 b=3 4 4 3 7 5 4 14 16 20 b=4 3 3 4 42 13 36 1 5 58 b=5 5 5 1 43 4 46 5 10 34 b=6 3 12 15 110 31 154 406 5 197 b=7 4 9 6 34 13 33 26 1440 104 b=8 6 6 45 84 41 249 171 6458 2801 b=9 9 30 14 52 16 74 20 14654 24 b=10 8 4 50 171 14 461 78 12203 312 b=11 4 14 30 10 36 332 666 16294 4686 b=12 5 10 26 116 39 603 120 6750 16105 b=13 7 9 61 57 9 263 130 13536 312 b=14 8 30 41 70 83 466 84 20008 16578 b=15 11 15 8 209 31 249 1010 31320 16806 b=16 9 12 54 224 25 666 312 19107 26294 s=11 s=12 s=13 s=14 s=15 s=16 b=2 6 5 7 7 5 6 b=3 1 1 11 6 6 8 b=4 5 6 7 161 8 70 b=5 6 12 374 12 409 7 b=6 101 396 7 937 311 968 b=7 36 249 753 235 1478 794 b=8 687 88 676 35 3129 2533 b=9 2211 28 28 32 32 1093 b=10 318880 2184 57725 5804 1401 9722 b=11 220110 1456 3666650 99291 8 69188 b=12 9076 354312 576360 161050 1044670 26256 b=13 188448 728 23487352 1456 21972045 720 b=14 1541063 9520 226440 10248 big big b=15 1396913 2394 3281355 1098056 10309491 13680 b=16 1013143 1456 28640 852852 24039416 976560 "big" means no loop had been detected when I stopped the program after 44 million terms, so tail length plus twice the loop length is > 44000000. (Thanks to Bill Gosper for the loop detector in HAKMEM item 132.) We don't know the tail length, so loop length can still be anything. A new large tail-to-loop ratio, for s=15, b=11, is 33668/8 = 4208.5. The loop is 99, 99, 99, 99, 104, 99, 99, a3 (base 11; a=ten). The loop lengths that occur above look rather random, getting sparser as they get larger, except for 5 values: loop length = 171 for s,b = 5,10 and 8,8 loop length = 249 for s,b = 7,8 and 7,15 and 12,7 loop length = 312 for s,b = 8,16 and 10,10 and 10,13 loop length = 666 for s,b = 7,16 and 8,11 loop length = 1456 for s,b = 12,11 and 12,16 and 14,13 Are these replications mere coincidence? Or is there a simple mapping of the structure between loops of the same length? -- Mike