Collatz Trajectory Records

I've written about the Collatz Trajectory or the \(3x+1\) Problem previously in the following posts:

• The \(3x+1\) Problem on November 1st 2016
• The Collatz Conjecture Revisited on March 15th 2018
• The \(17x+1\) Map on March 18th 2018
• The Primelatz Conjecture on August 25th 2019
• The Esucarys Mapping on February 15th 2021
• Patterns in Collatz Trajectories on April 3rd 2021

The topic was drawn to my attention yet again when I discovered that the number associated with my diurnal age, 26623, was intimately linked the \(3x+1\) problem because of its link to two sequences in the OEIS, specifically:

A006877

In the '3x+1' problem, these values for the starting value set  new records for number of steps to reach 1.

1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, ...

The record set is 307 steps. The full trajectory is as follows:

26623, 79870, 39935, 119806, 59903, 179710, 89855, 269566, 134783, 404350, 202175, 606526, 303263, 909790, 454895, 1364686, 682343, 2047030, 1023515, 3070546, 1535273, 4605820, 2302910, 1151455, 3454366, 1727183, 5181550, 2590775, 7772326, 3886163, 11658490, 5829245, 17487736, 8743868, 4371934, 2185967, 6557902, 3278951, 9836854, 4918427, 14755282, 7377641, 22132924, 11066462, 5533231, 16599694, 8299847, 24899542, 12449771, 37349314, 18674657, 56023972, 28011986, 14005993, 42017980, 21008990, 10504495, 31513486, 15756743, 47270230, 23635115, 70905346, 35452673, 106358020, 53179010, 26589505, 79768516, 39884258, 19942129, 59826388, 29913194, 14956597, 44869792, 22434896, 11217448, 5608724, 2804362, 1402181, 4206544, 2103272, 1051636, 525818, 262909, 788728, 394364, 197182, 98591, 295774, 147887, 443662, 221831, 665494, 332747, 998242, 499121, 1497364, 748682, 374341, 1123024, 561512, 280756, 140378, 70189, 210568, 105284, 52642, 26321, 78964, 39482, 19741, 59224, 29612, 14806, 7403, 22210, 11105, 33316, 16658, 8329, 24988, 12494, 6247, 18742, 9371, 28114, 14057, 42172, 21086, 10543, 31630, 15815, 47446, 23723, 71170, 35585, 106756, 53378, 26689, 80068, 40034, 20017, 60052, 30026, 15013, 45040, 22520, 11260, 5630, 2815, 8446, 4223, 12670, 6335, 19006, 9503, 28510, 14255, 42766, 21383, 64150, 32075, 96226, 48113, 144340, 72170, 36085, 108256, 54128, 27064, 13532, 6766, 3383, 10150, 5075, 15226, 7613, 22840, 11420, 5710, 2855, 8566, 4283, 12850, 6425, 19276, 9638, 4819, 14458, 7229, 21688, 10844, 5422, 2711, 8134, 4067, 12202, 6101, 18304, 9152, 4576, 2288, 1144, 572, 286, 143, 430, 215, 646, 323, 970, 485, 1456, 728, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

The number marked in bold in the trajectory (106358020) above is the highest value attained and this too sets a new Collatz trajectory record:

A006884

In the '3x+1' problem, these values for the starting value set new records for highest point of trajectory before reaching 1.

1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, 1042431, 1212415, 1441407, 1875711, 1988859, 2643183, 2684647, 3041127, 3873535, 4637979, 5656191

Figure 1 shows a plot of the 307 values from 26623 to 1:

Figure 1