Pseudo-loops with cubes

According to the OEIS, the list L of the first 26 cubes is (https://oeis.org/A000578/b000578.txt):

1^st = 0

2^nd = 1

3^rd = 8

4^th = 27

5^th = 64

6^th = 125

7^th = 216

8^th = 343

9^th = 512

10^th = 729

11^th = 1000

12^th = 1331

13^th = 1728

14^th = 2197

15^th = 2744

16^th = 3375

17^th = 4096

18^th = 4913

19^th = 5832

20^th = 6859

21^st = 8000

22^nd = 9261

23^rd = 10648

24^th = 12167

25^th = 13824

26^th = 15625

Idea#1

We start the sequence S with a(1) = 1;

we decide that k is the largest cube that can fit into a(n);

we then concatenate k and a(n) – k and iterate.

 

S = 1, 10, 82, 6418, 5832586, 5832000586, 5832000000586, 5832000000000586, ... 

Explanation

The largest cube that can fit into 1 is 1; 1-1 = 0; the concatenation forms 10;

the largest cube that can fit into 10 is 8; 10-8 = 2; the concatenation forms 82;    

the largest cube that can fit into 82 is 64; 82-64 = 18; the concatenation forms 6418;

the largest cube that can fit into 6418 is 5832; 6418-5832 = 586; the concatenation forms 5832586;

the largest cube that can fit into 5832586 is 5832000; 5832586-5832000 = 586; the

concatenation forms 5832000586;

this pattern goes on to infinity.

 

Starting the sequence T with a(1) = 2 would produce:

T = 2, 11, 83, 6419, 5832587, 5832000587, 5832000000587, 5832000000000587, ...

Starting the sequence U with a(1) = 3 would produce:

U = 3, 12, 84, 6420, 5832588, 5832000588, 5832000000588, 5832000000000588, ...

…

Starting the sequence V with a(1) = 7 would produce:

V = 7, 16, 88, 6424, 5832592, 5832000592, 5832000000592, 5832000000000592,...

Starting the sequence W with a(1) = 8 would produce:

W = 8, 80, 6416, 5832584, 5832000584, 5832000000584, 5832000000000584, ...

…

Starting the sequence X with a(1) = 26 would produce:

X = 26, 818, 72989, 689214068, 688465387748681, 88300000000748681, ...

Starting the sequence Y with a(1) = 27 would produce:

Y = 27, 270, 21654, 196831971, 196122941709030, 196122941000000709030, ...

 

Question

Will a(1) always end in such an infinite pseudo-loop, no matter its value?

____________________

Next morning update#1

Jean-Marc Falcoz is affirmative: yes, all starts seem to enter into an infinite pseudo-loop.

[English Google-translation after the French original version]

JMF

> Je te joins une liste de départs entre 1 et 9 pour lesquels, même avec seulement 15 itérations, c’est évident. Avec des départs de suite aux alentours de 35000, c’est la même chose.

[I am attaching a list of departures between 1 and 9 for which, even with only 15 iterations, it is obvious. With consecutive departures around 35,000, it's the same thing.]

Idea #1 (starts from 1 to 9)

{1,10,82,6418,5832586,5832000586,5832000000586,5832000000000586,5832000000000000586,5832000000000000000586,5832000000000000000000586,5832000000000000000000000586,5832000000000000000000000000586,5832000000000000000000000000000586,5832000000000000000000000000000000586,5832000000000000000000000000000000000586}

{2,11,83,6419,5832587,5832000587,5832000000587,5832000000000587,5832000000000000587,5832000000000000000587,5832000000000000000000587,5832000000000000000000000587,5832000000000000000000000000587,5832000000000000000000000000000587,5832000000000000000000000000000000587,5832000000000000000000000000000000000587}

{3,12,84,6420,5832588,5832000588,5832000000588,5832000000000588,5832000000000000588,5832000000000000000588,5832000000000000000000588,5832000000000000000000000588,5832000000000000000000000000588,5832000000000000000000000000000588,5832000000000000000000000000000000588,5832000000000000000000000000000000000588}

{4,13,85,6421,5832589,5832000589,5832000000589,5832000000000589,5832000000000000589,5832000000000000000589,5832000000000000000000589,5832000000000000000000000589,5832000000000000000000000000589,5832000000000000000000000000000589,5832000000000000000000000000000000589,5832000000000000000000000000000000000589}

{5,14,86,6422,5832590,5832000590,5832000000590,5832000000000590,5832000000000000590,5832000000000000000590,5832000000000000000000590,5832000000000000000000000590,5832000000000000000000000000590,5832000000000000000000000000000590,5832000000000000000000000000000000590,5832000000000000000000000000000000000590}

{6,15,87,6423,5832591,5832000591,5832000000591,5832000000000591,5832000000000000591,5832000000000000000591,5832000000000000000000591,5832000000000000000000000591,5832000000000000000000000000591,5832000000000000000000000000000591,5832000000000000000000000000000000591,5832000000000000000000000000000000000591}

{7,16,88,6424,5832592,5832000592,5832000000592,5832000000000592,5832000000000000592,5832000000000000000592,5832000000000000000000592,5832000000000000000000000592,5832000000000000000000000000592,5832000000000000000000000000000592,5832000000000000000000000000000000592,5832000000000000000000000000000000000592}

{8,80,6416,5832584,5832000584,5832000000584,5832000000000584,5832000000000000584,5832000000000000000584,5832000000000000000000584,5832000000000000000000000584,5832000000000000000000000000584,5832000000000000000000000000000584,5832000000000000000000000000000000584,5832000000000000000000000000000000000584,5832000000000000000000000000000000000000584}

{9,81,6417,5832585,5832000585,5832000000585,5832000000000585,5832000000000000585,5832000000000000000585,5832000000000000000000585,5832000000000000000000000585,5832000000000000000000000000585,5832000000000000000000000000000585,5832000000000000000000000000000000585,5832000000000000000000000000000000000585,5832000000000000000000000000000000000000585}

Idea #1 (starts from 35672 to 35677)

{35672,327682904,327082769600135,327082769000000600135,327082769000000000000600135,327082769000000000000000000600135,327082769000000000000000000000000600135,327082769000000000000000000000000000000600135,327082769000000000000000000000000000000000000600135,327082769000000000000000000000000000000000000000000600135,327082769000000000000000000000000000000000000000000000000600135,327082769000000000000000000000000000000000000000000000000000000600135,327082769000000000000000000000000000000000000000000000000000000000000600135,327082769000000000000000000000000000000000000000000000000000000000000000000600135,327082769000000000000000000000000000000000000000000000000000000000000000000000000600135,327082769000000000000000000000000000000000000000000000000000000000000000000000000000000600135}

{35673,327682905,327082769600136,327082769000000600136,327082769000000000000600136,327082769000000000000000000600136,327082769000000000000000000000000600136,327082769000000000000000000000000000000600136,327082769000000000000000000000000000000000000600136,327082769000000000000000000000000000000000000000000600136,327082769000000000000000000000000000000000000000000000000600136,327082769000000000000000000000000000000000000000000000000000000600136,327082769000000000000000000000000000000000000000000000000000000000000600136,327082769000000000000000000000000000000000000000000000000000000000000000000600136,327082769000000000000000000000000000000000000000000000000000000000000000000000000600136,327082769000000000000000000000000000000000000000000000000000000000000000000000000000000600136}

{35674,327682906,327082769600137,327082769000000600137,327082769000000000000600137,327082769000000000000000000600137,327082769000000000000000000000000600137,327082769000000000000000000000000000000600137,327082769000000000000000000000000000000000000600137,327082769000000000000000000000000000000000000000000600137,327082769000000000000000000000000000000000000000000000000600137,327082769000000000000000000000000000000000000000000000000000000600137,327082769000000000000000000000000000000000000000000000000000000000000600137,327082769000000000000000000000000000000000000000000000000000000000000000000600137,327082769000000000000000000000000000000000000000000000000000000000000000000000000600137,327082769000000000000000000000000000000000000000000000000000000000000000000000000000000600137}

{35675,327682907,327082769600138,327082769000000600138,327082769000000000000600138,327082769000000000000000000600138,327082769000000000000000000000000600138,327082769000000000000000000000000000000600138,327082769000000000000000000000000000000000000600138,327082769000000000000000000000000000000000000000000600138,327082769000000000000000000000000000000000000000000000000600138,327082769000000000000000000000000000000000000000000000000000000600138,327082769000000000000000000000000000000000000000000000000000000000000600138,327082769000000000000000000000000000000000000000000000000000000000000000000600138,327082769000000000000000000000000000000000000000000000000000000000000000000000000600138,327082769000000000000000000000000000000000000000000000000000000000000000000000000000000600138}

{35676,327682908,327082769600139,327082769000000600139,327082769000000000000600139,327082769000000000000000000600139,327082769000000000000000000000000600139,327082769000000000000000000000000000000600139,327082769000000000000000000000000000000000000600139,327082769000000000000000000000000000000000000000000600139,327082769000000000000000000000000000000000000000000000000600139,327082769000000000000000000000000000000000000000000000000000000600139,327082769000000000000000000000000000000000000000000000000000000000000600139,327082769000000000000000000000000000000000000000000000000000000000000000000600139,327082769000000000000000000000000000000000000000000000000000000000000000000000000600139,327082769000000000000000000000000000000000000000000000000000000000000000000000000000000600139}

{35677,327682909,327082769600140,327082769000000600140,327082769000000000000600140,327082769000000000000000000600140,327082769000000000000000000000000600140,327082769000000000000000000000000000000600140,327082769000000000000000000000000000000000000600140,327082769000000000000000000000000000000000000000000600140,327082769000000000000000000000000000000000000000000000000600140,327082769000000000000000000000000000000000000000000000000000000600140,327082769000000000000000000000000000000000000000000000000000000000000600140,327082769000000000000000000000000000000000000000000000000000000000000000000600140,327082769000000000000000000000000000000000000000000000000000000000000000000000000600140,327082769000000000000000000000000000000000000000000000000000000000000000000000000000000600140}

(Dall-e creation)

Idea#2

We start the sequence A with a(1) = 1;

we decide that k is the index (in the above list L) of the largest cube q that can fit into a(n);

we then concatenate k and a(n) - q and iterate.

A = 1, 20, 312, 796, 1067, 1167, 11167, 23519, ...   

Explanation

The largest cube that can fit into a(1) = 1 is 1 which is the 2^nd cube of the above list;

as a(1) - 1 = 0 we get the concatenation 20;

the largest cube that can fit into a(2) = 20 is 8 which is the 3^rd cube of the above list;

as a(2) - 8 = 12 we get the concatenation 312;

the largest cube that can fit into a(3) = 312 is 216 which is the 7^th cube of the above list;

as a(3) - 216 = 96 we get the concatenation 796;

the largest cube that can fit into a(4) = 796 is 729 which is the 10^th cube of the above list;

as a(4) - 729 = 67 we get the concatenation 1067;

the largest cube that can fit into a(5) = 1067 is 1000 which is the 11^th cube of the above list;

as a(5) - 1000 = 67 we get the concatenation 1167;

the largest cube that can fit into a(6) = 1167 is 1000 which is the 11^th cube of the above list;

as a(6) - 1000 = 167 we get the concatenation 11167;

the largest cube that can fit into a(7) = 11167 is 10648 which is the 23^rd cube of the above list;

as a(7) - 10648 = 519 we get the concatenation 23519; etc.

 

Question

Will the sequence A also enter in a pseudo-loop at some point?

____________________

Update#2 (answer by JMF to the above question)

Idea#2

Pour la version #2, je pense que cela ne boucle pas.

En partant de 1, après 2000 itérations, voici ce qu’on obtient.

[For the version #2, I think it doesn't loop.

Starting from 1, after 2000 iterations, here is what we obtain]

A = {1, 20, 312, 796, 1067, 1167, 11167, 23519, 291567, 674071, 8815568, 20773752, 275202928, 651577928, 8672116032, 20556462568, 27408153149, 30161124774, 311322823846, 677870604413, 8785108970109, 206341213993972, 5909210434503401, 18079184927464401, 262458181752706408, 640256712888125033, 861890186679386664, 9516652010718343720, 21191497080987373928, 276728521590211011624, 651655481269383567247, 8669739199239371111975, 20543228760944509673892, 27387743859772418351404, 301429272234369907552628, 6704943812787134271656175, 18856672134973851095208175, 266167495126599385486618391, 643257719239919371305468159, 8632335991303968100904424967, 2051364232506200644660886576, 12706158244625901658416439809, 233348360210967314447953109008, 615651465859387827286018757615,  (...),

 20536156111248574886172542737318355620260110903113883155923066395253076274680881991065894622325433633032304681829142940606481511637887472066719638556144595671050176826349643002710308072428215153920602934477882263195272116553894625408160058774573447022731698915297435496426946654802053291904134384771943850399663081485576805422499914456757887521260532588830632937973469912312944865489856703879295027882292395743070040010678340831743106328049068772266759571094224861189052158602907990377771450437521613548424356049197551069161953699579076119548795327227693161121358534141739166702915103026671323879918521265553572455572242266910459913975673278084066906901605814837426611915798443677958661463212782355506871866112002207863102088893765388110378475299532919718033344962572991360122408568146960619568417074384516217679522400238062497263280363659900131928686426849714220792150503609062040859679889277354704705132924170981715516019670561540189287898626333370351148689090042032162763996547586601564710722710293240450922419660119699618847769352067311004728167584482431424962659882272628516010231643386191696 (...),

273845989920425374179981079221240078736643378013458685058936045670132862620785281630134044572786192758400518780877209829810567100445408815041805558953275874225169955237897887746344761839520265450467879962867461123398840418124064526893495347523453283461943846967382774914531023904519528322830477856497008609237210418635941406917447793902707230095921135712058698606785171569088105127445843091931790792070354908054711709913797277626544234262763675634151440656013840838605528771094800986976184404847437553044101114002599409676419688322103156596298683240576669062053833817296589968389375511247611707351922508392932099626451635469802013536421537665637838232039259966610702874666133761298787975722140195502532029353708827594967350550551388529793523667188698725267774533714784961696264450945328562636726626941346319760554033595320980620100396316340093322084881100964573986351109358691274297070607225719439354591745824935416903527543299856886543646263984546978977624487763669990307477171803110795024474738575130464215639864230949172849197394338619054432708541674176203008052612855704174155911448984723693392}

Giorgos Kalogeropoulos has confirmed the results of Jean-Marc Falcoz

GK

> Here are the first 100 terms (of the Idea #2): 

1,20,312,796,1067,1167,11167,23519,291567,674071,8815568,20773752,275202928,651577928,8672116032,20556462568,27408153149,30161124774,311322823846,677870604413,8785108970109,206341213993972,5909210434503401,18079184927464401,262458181752706408,640256712888125033,861890186679386664,9516652010718343720,21191497080987373928,276728521590211011624,651655481269383567247,8669739199239371111975,20543228760944509673892,27387743859772418351404,301429272234369907552628,6704943812787134271656175,18856672134973851095208175,266167495126599385486618391,643257719239919371305468159,8632335991303968100904424967,2051364232506200644660886576,12706158244625901658416439809,233348360210967314447953109008,615651465859387827286018757615,8507036692103700078306911870244,20413905594696576986178390307387,27330151178411788378981151725154,301217831742418015358322425326437,670337567889768077630362181991034,8751809445217533828439467469175183,2060784682795496290869192468954231,12725578644114152598400685154970231,23346718474287250794467846286389210,28580859311542140499858123903023666,305744340946169246814071179373615041,6736786888931080628582034871293386753,18886476229125431829371529455271742722,26630765197551147689851224314433669658,29862618204558727669466919001666732994,310248216755221890279848882208832347843,676970532079041147191618135251006018774,878058102640178932244645715571262018774,9575785700476194092860808707375467346899,21235287218608899494744094261209485052356,276918973544941043443858452693719024191199,651804825048279282047339444524392115319223,8670401131598514222361280153823704512635319,20543750411104530940212286028192207263862135,27387974197295915345389696096385819596820223,301430113225703219630189026670512924802171815,670495003043075420363015171280304821333362591,8752494540107241581611674138474369548475559524,20608384543738306496783662879376514643818157735,274166665361913112203750288542100378753542660735,649638193409434071855409551556226637375241980516,866078351912100571794590901187065439853632172452,9532037182527830208466485242059598565650925428663,21202898902361672849572800177945648983167693313952,276778115187959721586063401358268236974566363030341,651694290033765633574586305044860659726787381494013,8669910986359509415694933641082539814288878706224656,20543363285178230999291462004669136154812772158062544,273878021634450010154849584738655301521804847486711815,649410132820730918605115652707912927967172166328706602,86597699218152645961733061235496770255172452549042690,442420712861801448235648066636087282663442251730701067,761982768663214067585236712742445518046328677241861571,913373450234786618365580339004533505954973541268314458,9702480855632401502281683717457218950247940745591286509,21328530361399476346319357615739057494762075810376201372,277323694460933623318685307903364397510625011612045122204,65212221107084437173174280840564496996449047095162236508,402509681455507330746618844629339758331438757609665299892,7383440457431505363157940765000335174658460244405044201964,19472404936598413169430684977224223083181028225988437924332,269033604774352715271580769328268262532533165590884276804756,645558360940844763389643605340473582068818321420752338000003,8642615060531530943916548841994797456266795333161950800584331,20521781378424265453248192993019482838704134600091958169381040,273782080171082026256172600975866992125329410874902078790849665,649334292933015844914985947257715586162268457273293673524093168.

____________________

Merci beaucoup Jean-Marc and Giorgos for those massive and beautiful numbers!

(mais Dieu sait pourquoi la version #1 entre systématiquement dans une pseudo-loop ?!)

[but God knows why the version #1 does systematically enter into a pseudo-loop?!)

____________________

Late update #3

Giorgos K.

>> but God knows why the version #1 ...

In your main example you state: "... concatenate k and a(n) – k and iterate"

Let's also define g^3 = k. Since k is a perfect cube, g is the cubic root of k which is an integer.

We can find a math formula for the concatenation of k and a(n) – k:

k is  the floor of cubic root of n -> k=floor(n^(1/3))^3 and g=floor(n^(1/3))

conc(n) = k (10^floor(log10(n-k)+1) -1) + n

Let's see an example...

for n=2 we have the following steps

step ->  g -> k             -> conc(n)

1->      1 -> 1             -> 11

2->      2 -> 8             -> 83
3->      4 -> 64            -> 6419
4->     18 -> 5832          -> 5832587
5->    180 -> 5832000       -> 5832000587
6->   1800 -> 5832000000    -> 5832000000587
7->  18000 -> 5832000000000 -> 5832000000000587

for n=3430 we have

1-> 15     -> 3375              -> 337555
2-> 69     -> 328509            -> 3285099046
3-> 1486   -> 3281379256        -> 32813792563719790
4-> 320148 -> 32813486631081792 -> 32813486631081792305932637998
5-> 3201480000

    -> 32813486631081792000000000000

    -> 32813486631081792000000000000305932637998
6-> 32014800000000

    -> 32813486631081792000000000000000000000000

    -> 32813486631081792000000000000000000000000305932637998
7-> 320148000000000000

    -> 32813486631081792000000000000000000000000000000000000

    -> 32813486631081792000000000000000000000000000000000000305932637998

We see that when g becomes a multiple of 10 we have the beginning of the loop.

So, can we find more steps? Can we delay g to become a multiple of 10?

A key number is n = 35947

For 35947 I will only display g because the other numbers become really massive

step -> g

 1 -> 33
 2 -> 153
 3 -> 7101
 4 -> 3295992
 5 -> 71009995028
 6 -> 3295991999968637484
 7 -> 7100999502799999999411777527165
 8 -> 3295991999968637483999999999999799629599274475136506
 9 -> 152986396627314784230928702562010852256091074332191360779455972328809471495295113233814
10 -> 710099950279999999941177752716499999999999999999999870212146323353281413060193308360313537897108587880027917179248547920072903345883937815964329
11 -> 329599199996863748399999999999979962959927447513650599999999999999999999999999999999999375778635863834198793694511014464563792083304748930726970995944373254670346474581218524966034673812333397515433709133037976878166239674430860351841650503
12 -> 1529863966273147842309287025620108522560910743321913607794559723288094714952951132338139999999999999999999999999999999999999999999999999999999999862390487487657231407387065186476250513113128412438327967555511338799036721473953151011986951325837333796934023322118297773762870483118655860126143318153015177779579726826078927358895886969623993380865030096499385476788481307898698210471859555072588711633

We see that after 12 steps we still do not have g divisible by 10.

In fact we need 20 steps to find such a g

[Because the numbers are very big, I will now display only the last 10 digits of g for the first 24 steps]

 1 -> 33
 2 -> 153
 3 -> 7101
 4 -> 3295992
 5 -> 1009995028
 6 -> 9968637484
 7 -> 1777527165
 8 -> 4475136506
 9 -> 5113233814
10 -> 7815964329
11 -> 1841650503
12 -> 2588711633
13 -> 2409950498
14 -> 2257697481
15 -> 4041182248
16 -> 7427602888
17 -> 4349477866
18 -> 6933425789
19 -> 5062525736
20 -> 1743438170 -> in this step k is divisible by 10
21 -> 603288493
22 -> 8752933169
23 -> 1733572718
24 -> 3173042015

To our surprise, g in the 21st step is not divisible by 10.

Why is that happening?

Does this mean we do not have a loop?

____________________

ÉA

Many thanks for this interesting post, Giorgos!

The mystery thickens!

(Dall-e creation)