The Collatz conjecture is a little math game, also known as the 3n+1 problem. Take a number, any number above zero and if it’s even, divide it by two; if it’s odd, multiply it by three and add one. Continue to do this until the value you started with ends at one. The conjecture is whether or not every value will reach one.
Various numbers can reach pretty high in their travels before finally going down to one. How long it takes to reach one are the number of steps or its stopping time. Here, for example, is the path 27 takes to reach one:
And here is 837,799:
The reason it stops at one is that one is the start of a loop. One going through the 3n+1 problem goes from 1 to 4 to 2 to 1. As of 2017, all values under 100 quintillion have been checked that they reach one. A lower bound on another loop, if one exists, requires at least 533 million steps.
But anyway, here is what plotting a number verus it’s stopping time looks like:
This pattern keeps growing outwards and onwards, not really constrained. When I first learned about this problem, I immediately took a look at the number of odd operations to even operations instead. That is, count how many times you divided by two (the even operation) and how many times you multiplied by three (the odd operation). Then divide the odd by the even and look at that. And this is what that looks like:
And it looks constrained! Here’s the first 25 million:
A further look out, it appears to converge towards 0.5ish:
Although, there are some starting values that still generate large o/e ratios:
The o/e ratio for the one loop is 0.5. Would other loops have the same value? Well, extending the 3n+1 problem to include negative values, you can find three more loops: -17 loop, -5 loop and -1 loop. Their o/e ratios are 7/11 (≈0.63636), 2/3 (≈0.66666) and 1.
And here are the maximum o/e ratios I’ve found:
Starting Value odd/even ratio 2 0 3 0.4 7 0.454545454545 9 0.461538461538 27 0.585714285714 230631 0.589928057554 626331 0.592476489028 837799 0.592705167173 1723519 0.593123209169 3732423 0.593582887701 5649499 0.59375 6649279 0.596153846154 8400511 0.596736596737 63728127 0.603040540541 3743559068799 0.603519668737
I stopped searching after 11 trillion. But these values are all a subset of the maximum steps, and if it’s true all of these would be a subset of that, then the next value would be 100759293214567 with a o/e ratio of 0.604938271605.
Speaking of the max step values, there’s a pattern that appears in their odd and even operations:
Starting Value odd/even 1 0/0 2 0/1 3 2/5 6 2/6 7 5/11 9 6/13 18 6/14 25 7/16 27 41/70 54 41/71 73 42/73 97 43/75 129 44/77 171 45/79 231 46/81 313 47/83 327 52/91 649 52/92 703 62/108 871 65/113 1161 66/115 2223 66/116 2463 76/132 2919 79/137 3711 87/150 6171 96/165 10971 98/169 13255 101/174 17647 102/176 23529 103/178 26623 113/194 34239 114/196 35655 119/204 52527 125/214 77031 129/221 106239 130/223 142587 138/236 156159 141/241 216367 142/243 230631 164/278 410011 166/282 511935 174/295 626331 189/319 837799 195/329 1117065 196/331 1501353 197/333 1723519 207/349 2298025 208/351 3064033 209/353 3542887 217/366 3732423 222/374 5649499 228/384 6649279 248/416 8400511 256/429 11200681 257/431 14934241 258/433 15733191 263/441 31466382 263/442 36791535 278/466 63728127 357/592 127456254 357/593 169941673 358/595 226588897 359/597 268549803 362/602 537099606 362/603 670617279 370/616 1341234558 370/617 1412987847 375/625 1674652263 378/630 2610744987 394/656 4578853915 408/679 4890328815 425/706 9780657630 425/707 12212032815 433/720 12235060455 445/739 13371194527 455/755 17828259369 456/757 31694683323 458/761 63389366646 458/762 75128138247 461/767 133561134663 463/771 158294678119 466/776 166763117679 471/784 202485402111 491/816 404970804222 491/817 426635908975 496/825 568847878633 497/827 674190078379 500/832 881715740415 501/834 989345275647 506/842 1122382791663 509/847 1444338092271 529/879 1899148184679 530/881 2081751768559 540/897 2775669024745 541/899 3700892032993 542/901 3743559068799 583/966 7487118137598 583/967 7887663552367 588/975 10516884736489 589/977 14022512981985 590/979 19536224150271 596/989 26262557464201 597/991 27667550250351 602/999 38903934249727 608/1009 48575069253735 616/1022 51173735510107 621/1030 60650353197163 624/1035 80867137596217 625/1037 100759293214567 686/1134 134345724286089 687/1136 223656998090055 696/1151 397612441048987 698/1155 530149921398649 699/1157 706866561864865 700/1159 942488749153153 701/1161 1256651665537537 702/1163 1675535554050049 703/1165 2234047405400065 704/1167 2978729873866753 705/1169 3586720916237671 713/1182 4320515538764287 716/1187 4861718551722727 721/1195 6482291402296969 722/1197 7579309213675935 737/1221 12769884180266527 768/1271 17026512240355369 769/1273 22702016320473825 770/1275 45404032640947650 770/1276 46785696846401151 787/1303
The pattern isn’t completely there, but for the next max step, the odd operation count goes up by one and the even operation step goes up by two. You can see one of the breaks in the pattern near the end of the above list, where the even count goes up by one instead of two. It makes sense, since for every one odd operations, you’ll need two even operations to counter it – curious how it only needs one additional set of the operations to hit the next maximum step though.
In any case? It’s just a different look at it.
Orthallelous 
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