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 14 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 second 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.
Back to the general 3x+1 page.
|
|
#
|
Delay
|
N
|
Level
|
Residue
|
Remark
|
|---|
|
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
|
|