Upward and downward steps in Collatz orbits started at k returning to k+-1, sorted by orbit lengths. # indicates upward 3*x+1 steps . indicates downward x/2 steps first column: k (terms in OEIS A355239) 2 orbits with 3 steps: 5 |#..| 4 6 |.#.| 5 4 orbits with 8 steps: 11 |#.#..#..| 10 14 |.#.#.#..| 13 17 |#..#...#| 16 18 |.#..#.#.| 17 13 orbits with 13 steps: 7 |#.#.#..#...#.| 8 9 |#..#.#.#..#..| 10 15 |#.#.#.#.....#| 16 19 |#.#...#.#..#.| 20 25 |#..#.#...#.#.| 26 33 |#..#..#.#...#| 34 39 |#.#.#..#.#...| 38 51 |#.#...#...#.#| 52 57 |#..#.#..#..#.| 56 59 |#.#..#.#....#| 58 81 |#..#...#.#.#.| 80 89 |#..#.#....#.#| 88 108 |..#.#..#.#.#.| 107 10 orbits with 44 steps: 159 |#.#.#.#.#..#.#.#.#.#....#..#..#....#...#.#.#| 160 166 |.#.#...#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...| 167 250 |.#...#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.| 251 252 |..#.#.#.#.#.#....#.#..#.#.#..#.#.#.#..#...#.| 251 284 |..#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..| 283 333 |#...#...#.#.#.#..#..#.#..#.#.#..#.#.#.#..#..| 334 376 |...#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.| 377 378 |.#...#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.| 377 411 |#.#..#.#.#.#...#.#..#......#.#.#.#.#..#..#.#| 412 501 |#.....#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#| 502 19 orbits with 75 steps: 91 |#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#.| 92 121 |#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#...| 122 183 |#.#.#...#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#| 184 243 |#.#...#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#..| 244 639 |#.#.#.#.#.#.#..#.#..#.#.#..#...#.....#...#..#.#.#..#.#.#.#..#...#.#.#..#.#.| 638 651 |#.#..#...#.#...#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#.| 650 667 |#.#..#.#.#.#..#.#.#.#..#.#...#.#.#.......#.#.#.#....#.#..#.#.#..#.#.#.#..#.| 668 889 |#..#.#..#.#.#.#..#.#.#.#..#.#...#.#.#.......#.#.#.#....#.#..#.#.#..#.#.#.#.| 890 959 |#.#.#.#.#.#..#.#..#.#.#..#...#.....#...#..#.#.#..#.#.#.#..#...#.#.#..#.#..#| 958 977 |#..#...#.#...#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#| 976 1185 |#..#..#.#..#.#.#.#..#.#.#.#..#.#...#.#.#.......#.#.#.#....#.#..#.#.#..#.#.#| 1186 1335 |#.#.#...#.#.#..#.#.#.#..#.#...#.#.#.......#.#.#.#....#.#..#.#.#..#.#.#.#..#| 1336 1439 |#.#.#.#.#..#.#..#.#.#..#...#.....#...#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#| 1438 1779 |#.#...#..#.#.#.#..#.#.#.#..#.#...#.#.#.......#.#.#.#....#.#..#.#.#..#.#.#.#| 1780 2159 |#.#.#.#..#.#..#.#.#..#...#.....#...#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#| 2158 3239 |#.#.#..#.#..#.#.#..#...#.....#...#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#| 3238 4617 |#..#.#.#..#.......#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#.| 4616 4859 |#.#..#.#..#.#.#..#...#.....#...#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#| 4858 7289 |#..#.#..#.#.#..#...#.....#...#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#| 7288 7 orbits with 88 steps: 47 |#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#..| 46 62 |.#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....| 61 71 |#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#...#| 70 107 |#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#...#.#| 106 161 |#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#...#.#.#| 160 432 |....#.#..#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..| 433 1368 |...#.#..#..#..#...#..#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.| 1367 2 orbits with 101 steps: 41 |#..#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#...#.#.#..| 40 54 |.#.#..#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#....#...#.#.| 53 17 orbits with 119 steps: 487 |#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#.| 488 649 |#..#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#.| 650 865 |#..#..#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#.| 866 975 |#.#.#.#...#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#..#| 976 1153 |#..#..#..#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#...| 1154 1299 |#.#...#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#..#| 1300 1731 |#.#....#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#....#| 1732 1823 |#.#.#.#.#..#...#....#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#..| 1822 2307 |#.#....#..#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#..| 2308 2430 |.#.#.#.#.#.#..#...#....#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.| 2429 2735 |#.#.#.#..#...#....#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#| 2734 3643 |#.#..#.#....#..#..#..#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#.| 3644 4103 |#.#.#..#...#....#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#| 4102 4857 |#..#.#..#.#....#..#..#..#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#| 4858 6155 |#.#..#...#....#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#| 6154 7287 |#.#.#...#....#..#..#..#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#| 7288 9233 |#..#...#....#.#.#..#.#.#...#.#.#...#.#..#......#.#.#.#.#..#..#.#..#.#.#..#.#.#.#..#...#.#.#..#.#..#.#.#.#.#.#...#.#.#.#| 9232 Hugo Pfoertner http://www.pfoertner.org 2022-07-15