Research Article
On the Probabilistic Proof of the Convergence of the Collatz Conjecture
Table 1
Positive even integers and their corresponding frequencies of division by 2. Any row has the same frequency. Even elements of
function are in italic.
| | # | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. | F | Int. |
| | 2s | 3 | 8 | 4 | 16 | 5 | 32 | 6 | 64 | 7 | 128 | 8 | 256 | 9 | 512 | 10 | 1024 | 30 | 1073741824 |
| | 2 | 1 | 10 | 1 | 18 | 1 | 34 | 1 | 66 | 1 | 130 | 1 | 258 | 1 | 514 | 1 | 1026 | 1 | 1073741826 |
| | 3 | 2 | 12 | 2 | 20 | 2 | 36 | 2 | 68 | 2 | 132 | 2 | 260 | 2 | 516 | 2 | 1028 | 2 | 1073741828 |
| | 4 | 1 | 14 | 1 | 22 | 1 | 38 | 1 | 70 | 1 | 134 | 1 | 262 | 1 | 518 | 1 | 1030 | 1 | 1073741830 |
| | 5 | | | 3 | 24 | 3 | 40 | 3 | 72 | 3 | 136 | 3 | 264 | 3 | 520 | 3 | 1032 | 3 | 1073741832 |
| | 6 | | | 1 | 26 | 1 | 42 | 1 | 74 | 1 | 138 | 1 | 266 | 1 | 522 | 1 | 1034 | 1 | 1073741834 |
| | 7 | | | 2 | 28 | 2 | 44 | 2 | 76 | 2 | 140 | 2 | 268 | 2 | 524 | 2 | 1036 | 2 | 1073741836 |
| | 8 | | | 1 | 30 | 1 | 46 | 1 | 78 | 1 | 142 | 1 | 270 | 1 | 526 | 1 | 1038 | 1 | 1073741838 |
| | 9 | | | | | 4 | 48 | 4 | 80 | 4 | 144 | 4 | 272 | 4 | 528 | 4 | 1040 | 4 | 1073741840 |
| | 10 | | | | | 1 | 50 | 1 | 82 | 1 | 146 | 1 | 274 | 1 | 530 | 1 | 1042 | 1 | 1073741842 |
| | 11 | | | | | 2 | 52 | 2 | 84 | 2 | 148 | 2 | 276 | 2 | 532 | 2 | 1044 | 2 | 1073741844 |
| | 12 | | | | | 1 | 54 | 1 | 86 | 1 | 150 | 1 | 278 | 1 | 534 | 1 | 1046 | 1 | 1073741846 |
| | 13 | | | | | 3 | 56 | 3 | 88 | 3 | 152 | 3 | 280 | 3 | 536 | 3 | 1048 | 3 | 1073741848 |
| | 14 | | | | | 1 | 58 | 1 | 90 | 1 | 154 | 1 | 282 | 1 | 538 | 1 | 1050 | 1 | 1073741850 |
| | 15 | | | | | 2 | 60 | 2 | 92 | 2 | 156 | 2 | 284 | 2 | 540 | 2 | 1052 | 2 | 1073741852 |
| | 16 | | | | | 1 | 62 | 1 | 94 | 1 | 158 | 1 | 286 | 1 | 542 | 1 | 1054 | 1 | 1073741854 |
| | 17 | | | | | | | 5 | 96 | 5 | 160 | 5 | 288 | 5 | 544 | 5 | 1056 | 5 | 1073741856 |
| | 18 | | | | | | | 1 | 98 | 1 | 162 | 1 | 290 | 1 | 546 | 1 | 1058 | 1 | 1073741858 |
| | 19 | | | | | | | 2 | 100 | 2 | 164 | 2 | 292 | 2 | 548 | 2 | 1060 | 2 | 1073741860 |
| | 20 | | | | | | | 1 | 102 | 1 | 166 | 1 | 294 | 1 | 550 | 1 | 1062 | 1 | 1073741862 |
| | 21 | | | | | | | 3 | 104 | 3 | 168 | 3 | 296 | 3 | 552 | 3 | 1064 | 3 | 1073741864 |
| | 22 | | | | | | | 1 | 106 | 1 | 170 | 1 | 298 | 1 | 554 | 1 | 1066 | 1 | 1073741866 |
| | 23 | | | | | | | 2 | 108 | 2 | 172 | 2 | 300 | 2 | 556 | 2 | 1068 | 2 | 1073741868 |
| | 24 | | | | | | | 1 | 110 | 1 | 174 | 1 | 302 | 1 | 558 | 1 | 1070 | 1 | 1073741870 |
| | 25 | | | | | | | 4 | 112 | 4 | 176 | 4 | 304 | 4 | 560 | 4 | 1072 | 4 | 1073741872 |
| | 26 | | | | | | | 1 | 114 | 1 | 178 | 1 | 306 | 1 | 562 | 1 | 1074 | 1 | 1073741874 |
| | 27 | | | | | | | 2 | 116 | 2 | 180 | 2 | 308 | 2 | 564 | 2 | 1076 | 2 | 1073741876 |
| | 28 | | | | | | | 1 | 118 | 1 | 182 | 1 | 310 | 1 | 566 | 1 | 1078 | 1 | 1073741878 |
| | 29 | | | | | | | 3 | 120 | 3 | 184 | 3 | 312 | 3 | 568 | 3 | 1080 | 3 | 1073741880 |
| | 30 | | | | | | | 1 | 122 | 1 | 186 | 1 | 314 | 1 | 570 | 1 | 1082 | 1 | 1073741882 |
| | 31 | | | | | | | 2 | 124 | 2 | 188 | 2 | 316 | 2 | 572 | 2 | 1084 | 2 | 1073741884 |
| | 32 | | | | | | | 1 | 126 | 1 | 190 | 1 | 318 | 1 | 574 | 1 | 1086 | 1 | 1073741886 |
| | 33 | | | | | | | | | 6 | 192 | 6 | 320 | 6 | 576 | 6 | 1088 | 6 | 1073741888 |
| | 34 | | | | | | | | | 1 | 194 | 1 | 322 | 1 | 578 | 1 | 1090 | 1 | 1073741890 |
| | 35 | | | | | | | | | 2 | 196 | 2 | 324 | 2 | 580 | 2 | 1092 | 2 | 1073741892 |
| | 36 | | | | | | | | | 1 | 198 | 1 | 326 | 1 | 582 | 1 | 1094 | 1 | 1073741894 |
| | 37 | | | | | | | | | 3 | 200 | 3 | 328 | 3 | 584 | 3 | 1096 | 3 | 1073741896 |
| | 38 | | | | | | | | | 1 | 202 | 1 | 330 | 1 | 586 | 1 | 1098 | 1 | 1073741898 |
| | 39 | | | | | | | | | 2 | 204 | 2 | 332 | 2 | 588 | 2 | 1100 | 2 | 1073741900 |
| | 40 | | | | | | | | | 1 | 206 | 1 | 334 | 1 | 590 | 1 | 1102 | 1 | 1073741902 |
| | 41 | | | | | | | | | 4 | 208 | 4 | 336 | 4 | 592 | 4 | 1104 | 4 | 1073741904 |
| | 42 | | | | | | | | | 1 | 210 | 1 | 338 | 1 | 594 | 1 | 1106 | 1 | 1073741906 |
| | 43 | | | | | | | | | 2 | 212 | 2 | 340 | 2 | 596 | 2 | 1108 | 2 | 1073741908 |
| | 44 | | | | | | | | | 1 | 214 | 1 | 342 | 1 | 598 | 1 | 1110 | 1 | 1073741910 |
| | 45 | | | | | | | | | 3 | 216 | 3 | 344 | 3 | 600 | 3 | 1112 | 3 | 1073741912 |
| | 46 | | | | | | | | | 1 | 218 | 1 | 346 | 1 | 602 | 1 | 1114 | 1 | 1073741914 |
| | 47 | | | | | | | | | 2 | 220 | 2 | 348 | 2 | 604 | 2 | 1116 | 2 | 1073741916 |
| | 48 | | | | | | | | | 1 | 222 | 1 | 350 | 1 | 606 | 1 | 1118 | 1 | 1073741918 |
| | 49 | | | | | | | | | 5 | 224 | 5 | 352 | 5 | 608 | 5 | 1120 | 5 | 1073741920 |
| | 50 | | | | | | | | | 1 | 226 | 1 | 354 | 1 | 610 | 1 | 1122 | 1 | 1073741922 |
| | 51 | | | | | | | | | 2 | 228 | 2 | 356 | 2 | 612 | 2 | 1124 | 2 | 1073741924 |
| | 52 | | | | | | | | | 1 | 230 | 1 | 358 | 1 | 614 | 1 | 1126 | 1 | 1073741926 |
| | 53 | | | | | | | | | 3 | 232 | 3 | 360 | 3 | 616 | 3 | 1128 | 3 | 1073741928 |
| | 54 | | | | | | | | | 1 | 234 | 1 | 362 | 1 | 618 | 1 | 1130 | 1 | 1073741930 |
| | 55 | | | | | | | | | 2 | 236 | 2 | 364 | 2 | 620 | 2 | 1132 | 2 | 1073741932 |
| | 56 | | | | | | | | | 1 | 238 | 1 | 366 | 1 | 622 | 1 | 1134 | 1 | 1073741934 |
| | 57 | | | | | | | | | 4 | 240 | 4 | 368 | 4 | 624 | 4 | 1136 | 4 | 1073741936 |
| | 58 | | | | | | | | | 1 | 242 | 1 | 370 | 1 | 626 | 1 | 1138 | 1 | 1073741938 |
| | 59 | | | | | | | | | 2 | 244 | 2 | 372 | 2 | 628 | 2 | 1140 | 2 | 1073741940 |
| | 60 | | | | | | | | | 1 | 246 | 1 | 374 | 1 | 630 | 1 | 1142 | 1 | 1073741942 |
| | 61 | | | | | | | | | 3 | 248 | 3 | 376 | 3 | 632 | 3 | 1144 | 3 | 1073741944 |
| | 62 | | | | | | | | | 1 | 250 | 1 | 378 | 1 | 634 | 1 | 1146 | 1 | 1073741946 |
| | 63 | | | | | | | | | 2 | 252 | 2 | 380 | 2 | 636 | 2 | 1148 | 2 | 1073741948 |
| | 64 | | | | | | | | | 1 | 254 | 1 | 382 | 1 | 638 | 1 | 1150 | 1 | 1073741950 |
| | 65 | | | | | | | | | | | 7 | 384 | 7 | 640 | 7 | 1152 | 7 | 1073741952 |
| | | | | | | | | | | | | .. | ... | … | … | … | … | … | … |
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