3x+1 Delay Records

The table on the right depicts all currently known and confirmed delay records. Numbers with much higher delays are obviously known, but of course a record can only be considered a confirmed record when all smaller numbers have been checked and found to have a lower Delay. Records up to number 106 were already published in 1992 by Leavens and Vermeulen. As far as the author is aware the others were all found (or confirmed) by the distributed project.

From these data a few interesting observations can be made. First of all it is noteworthy that approximately eight records per decade (power of ten) are found, implying that on average a delay record is 1.33 times larger then its predecessor. It should be obvious that the maximum gap that can occur is a factor two, when a new record is exactly twice the previous record. This is no rare occcurrence, and indeed no less than 15 delay records are 'doubles', improving the previous record by just one. Four of these are less than 100, when candidates for new records are rare anyway. Most others occur when a previous record has yielded a big improvement. The highest currently known record is also a double.

Many improvements in delay are just small. In a few cases though a delay record hugely improves the previous record. Record #59, at 63,728,127 improves the previous record by 205, easily the largest gap found so far. The only three other gaps to exceed 100 are 106, by record #96, 158 by record #109 and 163 by record #131. Interestingly these numbers are the lowest numbers of levels -1, -2, -3 and -4 respectively. Also all four are among the extremely rare strength records, of which only five are known with certainty.

The 'residue' column shows that many delay records are in a way 'related', sharing near identical residues. Record #59 shares its residue of 1.183418 with the next 11 delay records, showing that their paths are coalescing well before they reach any numbers below 1,000,000. Similarly 11 delay records share the residue 1.151611. Beyond that another 'family' takes over, all sharing their residue of 1.148153 with strength record number 3. Finally no less than 16 consecutive delay records, among which we find strength record 4, show a residue of 1.146963.


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# Delay N Level  Residue  Remark
132 2255 209798,591621,802462 -4 1.181068 Double
131 2254 104899,295810,901231 -4 1.181068 Strength record #5
130 2091 93571,393692,802302 -2 1.195484 Double
129 2090 46785,696846,401151 -3 1.195484  
128 2046 45404,032640,947650 -2 1.185261 Double
127 2045 22702,016320,473825 -3 1.185261  
126 2042 17026,512240,355369 -3 1.185261  
125 2039 12769,884180,266527 -3 1.185261  
124 1958 7579,309213,675935 -2 1.095545  
123 1919 6482,291402,296969 -2 1.095545  
122 1916 4861,718551,722727 -2 1.095545  
121 1903 4320,515538,764287 -2 1.170175  
120 1895 3586,720916,237671 -2 1.189331  
119 1874 2978,729873,866753 -2 1.146963  
118 1871 2234,047405,400065 -2 1.146963  
117 1868 1675,535554,050049 -2 1.146963  
116 1865 1256,651665,537537 -2 1.146963  
115 1862 942,488749,153153 -2 1.146963  
114 1859 706,866561,864865 -2 1.146963  
113 1856 530,149921,398649 -3 1.146963  
112 1853 397,612441,048987 -3 1.146963  
111 1847 223,656998,090055 -3 1.146963  
110 1823 134,345724,286089 -3 1.146963  
109 1820 100,759293,214567 -3 1.146963 Strength record #4
108 1662 80,867137,596217 -1 1.146963  
107 1659 60,650353,197163 -1 1.146963  
106 1651 51,173735,510107 -1 1.146963  
105 1638 48,575069,253735 -1 1.146963  
104 1617 38,903934,249727 -1 1.146963  
103 1601 27,667550,250351 -1 1.148153  
102 1588 26,262557,464201 -1 1.148153  
101 1585 19,536224,150271 -1 1.157597  
100 1569 14,022512,981985 -1 1.148153  
99 1566 10,516884,736489 -1 1.148153  
98 1563 7,887663,552367 -1 1.148153  
97 1550 7,487118,137598 -1 1.148153 Double
96 1549 3,743559,068799 -2 1.148153 Strength record #3
95 1443 3,700892,032993 0 1.148153  
94 1440 2,775669,024745 -1 1.148153  
93 1437 2,081751,768559 -1 1.148153  
92 1411 1,899148,184679 0 1.133972  
91 1408 1,444338,092271 -1 1.118288  
90 1356 1,122382,791663 0 1.168279  
89 1348 989345,275647 0 1.118288  
88 1335 881715,740415 0 1.191075  
87 1332 674190,078379 0 1.168279  
86 1324 568847,878633 0 1.168279  
85 1321 426635,908975 0 1.168279  
84 1308 404970,804222 0 1.168279 Double
83 1307 202485,402111 0 1.168279  
82 1255 166763,117679 0 1.151611  
81 1242 158294,678119 0 1.151611  
80 1234 133561,134663 0 1.151611  
79 1228 75128,138247 0 1.151611  
78 1220 63389,366646 0 1.151611 Double
77 1219 31694,683323 0 1.151611  
76 1213 17828,259369 -1 1.151611  
75 1210 13371,194527 -1 1.151611 Strength record #2
74 1184 12235,060455 -1 1.133972  
73 1153 12212,032815 0 1.151611  
72 1132 9780,657630 0 1.151611 Double
71 1131 4890,328815 0 1.151611  
70 1087 4578,853915 0 1.183418  
69 1050 2610,744987 0 1.183418  
68 1008 1674,652263 0 1.183418  
67 1000 1412,987847 0 1.183418  
66 987 1341,234558 1 1.183418 Double
65 986 670,617279 0 1.183418  
64 965 537,099606 0 1.183418 Double
63 964 268,549803 0 1.183418  
62 956 226,588897 0 1.183418  
61 953 169,941673 0 1.183418  
60 950 127,456254 0 1.183418 Double
59 949 63,728127 -1 1.183418 Strength record #1
58 744 36,791535 1 1.187193  
57 705 31,466382 2 1.187193 Double
56 704 15,733191 1 1.187193  
55 691 14,934241 2 1.187193  
54 688 11,200681 1 1.187193  
53 685 8,400511 1 1.187193  
52 664 6,649279 1 1.201247  
51 612 5,649499 2 1.147790  
50 596 3,732423 2 1.236827  
49 583 3,542887 2 1.236827  
48 562 3,064033 2 1.145388  
47 559 2,298025 2 1.145388  
46 556 1,723519 2 1.145388  
45 530 1,501353 2 1.184728  
44 527 1,117065 2 1.194220  
43 524 837799 2 1.194219  
42 508 626331 2 1.137228  
41 469 511935 2 1.189969  
40 448 410011 2 1.189967  
39 442 230631 2 1.189968  
38 385 216367 3 1.158457  
37 382 156159 3 1.203830  
36 374 142587 3 1.112413  
35 353 106239 3 1.195754  
34 350 77031 3 1.236863  
33 339 52527 3 1.147835  
32 323 35655 3 1.203842  
31 310 34239 3 1.189967  
30 307 26623 3 1.147785  
29 281 23529 3 1.170163  
28 278 17647 3 1.170147  
27 275 13255 3 1.168403  
26 267 10971 3 1.191078  
25 261 6171 2 1.191114  
24 237 3711 2 1.189760  
23 216 2919 2 1.211424  
22 208 2463 2 1.211378  
21 182 2223 3 1.209309  
20 181 1161 2 1.157749  
19 178 871 2 1.157417  
18 170 703 2 1.209947  
17 144 649 2 1.180891  
16 143 327 2 1.171862  
15 130 313 2 1.162108  
14 127 231 2 1.180973  
13 124 171 2 1.196512  
12 121 129 2 1.189556  
11 118 97 2 1.186490  
10 115 73 2 1.182426  
9 112 54 1 1.198849 Double
8 111 27 1 1.198849  
7 23 25 2 1.198647  
6 20 18 2 1.248590 Double
5 19 9 2 1.248590  
4 16 7 1 1.203998  
3 8 6 1 1.185185 Double
2 7 3 1 1.185185  
1 1 2 1 1.000000 Double

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