# Path records

The Collatz conjecture asserts that a sequence defined by repeatedly applying the function $$T(n) = \begin{cases} (3n+1)/2 &\text{ if n is odd, or}\\ n/2 &\text{ if n is even} \end{cases}$$ will always converge to the cycle passing through the number 1 for arbitrary initial positive integer $n$. Define $t(n)$ the highest number occurring in the sequence starting at $n$. The $n$ is called the path record if for all $m < n$ the inequality $t(m) < t(n)$ holds. The following table lists the known path records for all $n$ below $1536 \times 2^{60}$. $B(n)$ and $B(t(n))$ are the sizes of their arguments measured in the number of bits. Lagarias and Weiss (1992) predicted using the theory for random walks that $$\limsup\limits_{n\rightarrow\infty} \frac{\log t(n)}{\log n} = 2 \text{.}$$ In other words, the highest number occurring in the sequence for a path record $n$ grows like $n^2$. This agrees with the empirical evidence given in the table below.

#$n$$t(n)$$B(n)$$B(t(n))$$\log t(n) / \log n$
011111.000000
122221.000000
238241.892789
3726351.674330
41580471.618148
52746165132.559982
625565608131.586054
7447196829151.620215
86392076210151.538859
970312525210171.790609
10181963846811201.780809
114255340506813221.800029
124591407681013221.805158
1396631355721214241.789704
14208952507163215251.712757
15266235317901015261.745829
16319116050643215261.727776
176097529663957616291.770525
187767178541236817301.818942
19113383124105567417311.799130
20138367139916168018311.778994
21159487860118887618341.909491
222702711232403894819341.857718
236652152624164265620351.789294
247045112849574176020351.787787
2510424314511957782420361.770402
2612124156982336840421371.782482
2714414077581478718621371.766542
2818757117795217484821371.736257
2919888597845718911221371.729690
3026431839522990924222371.709523
31268464717630890647222381.749335
32304112731135895081022391.772821
33387353542927758478822391.765718
34463797965940114746623401.772962
355656191120624680830423411.789173
366416623239999847268423421.818665
3766316753017130545981623451.976010
381963839915314846260187625481.945003
393859558323731884942554626481.894848
4080049391109257191458505027501.902792
41120080895163895078805929027511.883113
42210964383320239858056063228521.862697
4331980483170711822335997124029602.098734
441410123943356294256139722608031622.027685
458528817511907229746867829901233631.908965
46123278295031036119945720252586434641.884414
47230355374073441907832077411352035651.885355
48458719622714117082445101141700236661.839751
49517393364475731980857080699922036661.844188
50591526410557574968253119510077236671.845472
515943613566310286819468592092608436671.857452
527014125977521048355689419491485237681.873803
537756636255945830651453843389992837691.897316
5411024309427168622682478313419018037701.886959
5520443061324770763039650482749554438701.843395
56231913730799109517191194143725677838701.851199
572720256605431097424181783520898187438741.927514
584465592172791976663845538903019053639751.913834
595678398626315027008661279299311799440761.931332
6087167382844320027937041062506101686440781.951503
61267430954764738520997492487118652613642791.897907
62371650998819910396823167227497452243773242872.069739
63901634607051112611476359172166759721209644872.014721
646484822433714763705346010407923289313386446901.940658
65116050121715711126529203391689248061311819647911.926973
66201321227677935263697551208880300194698520848921.917038
67265078413377535285720407807896655584771682648921.903573
68291732129855135353755893613372676024332846449921.904510
69394491988532895605428211322744550460691965049931.903398
704067389209606671280069670502122841144261968249941.923925
716134501766625112288144174297286214599261977650951.917765
727374822360531193768466579878244669010750592850951.922023
7312542518747743751823036311464280263720932141024511012.004241
7453230482328132471964730439297455725829478995944531011.926300
75856223501402665513471057008351679202003944688336531041.953820
7610709980568908647175294593968539094415936960141122541082.011491
7749163256101584231301753104069007668258074264675786561081.945863
7882450591202377887875612750096198197075499421245450571101.947383
79932647925034581192115362774686865777485863406680032571111.963815
801725453311995106312118089541282012618909185268056780581111.933407
812125815587801413112176718166004315761101410771585688581111.923979
822558753361340000632415428612584587115646993931986234581111.917678
834845499931280972154332751846533208436890106993276834591121.901958
845623807584222542716718947974962862349115040884231904591131.905760
8562822628637475292331268160888027375005205169043314754601151.938132
86891563131061253151140246903347442029303138585287425762601171.958028
871038743969413717663159695671984678120932209599662553676601171.953946
88198097605769484844732012333661096566765082938647132369010611252.049820
893513622115866480025592102545196486820634779928066830214436651271.942328
9048503373501652785087296696710908147364747230298439288489642661281.954321
9155247846101001863167482192631091346876742345874501396316228661291.959406
92711493236741026244154527691962113372170289733115168874698466661321.997559
9327413305463235210626756649062372194325899121269007146717645316681361.993995
94137829970034363369149573161949456862481446077319071744298810624711361.933091
95173551916886591445127188638514583197544506368462565103733234728711371.927906
961765856170146672440559394170168671074358667552016954030019514164711391.957724

## References

• Lagarias, J. C.; Weiss, A. The $3x+1$ Problem: Two Stochastic Models. Ann. Appl. Probab. 2 (1992), no. 1, 229–261. doi:10.1214/aoap/1177005779.
• Sequence A006884 in The On-Line Encyclopedia of Integer Sequences.
• Eric Roosendaal, 3x + 1 Path Records.

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