Suite des K-phénix

Suite des K-phénix

On commence par un entier K > 9 (10 par exemple), on multiplie les deux chiffres les plus à gauche, on colle le résultat tout à droite derrière K, on efface le chiffre le plus à gauche, on itère (on trouvera dans l’OEIS le même protocole concernant l’addition — alors qu’il s’agit ici de multiplier à gauche).

Nous nous intéressons aux K-phénix qui réapparaissent dans la concaténation ainsi générée.

La suite des K-phénix semble commencer par :

K = 12, 21, 22, 23, 24, 26, 28, … (mais nous ne sommes sûrs de rien)

Aucun K-phénix ne se termine en 0 ou 5 (mais 53 est dans la suite K car 53->5315->3153).

100

1000

10000...

K = 10 finit en point fixe sur 0

111

1111

11111...

K = 11 finit en point fixe sur 1

122

1224

12248

1224832

122483224

1224832246

12248322464

122483224648

12248322464824

1224832246482424

122483224648242432

12248322464824243216

122483224648242432168

1224832246482424321688

12248322464824243216888

1224832246482424321688812 <— hit

212

2122

21224

212248

21224832

2122483224

21224832246

212248322464

2122483224648

212248322464824

21224832246482424

2122483224648242432

212248322464824243216 <— hit

224

2248

224832

22483224 <— hit

236

23618

236186

2361868

236186848

23618684848

2361868484832

236186848483232 <— hit

248

24832

2483224 <— hit

2612

26126 <— hit

2816

28168

281686

28168648

2816864848

281686484824

28168648482432

2816864848243232

281686484824323232

28168648482432323216

281686484824323232168

28168648482432323216812

281686484824323232168126

2816864848243232321681266

28168648482432323216812666

281686484824323232168126666

2816864848243232321681266666

28168648482432323216812666662

281686484824323232168126666626

28168648482432323216812666662648

281686484824323232168126666626488

2816864848243232321681266666264882

281686484824323232168126666626488212

28168648482432323216812666662648821236

2816864848243232321681266666264882123636

281686484824323232168126666626488212363636

28168648482432323216812666662648821236363636

2816864848243232321681266666264882123636363612

281686484824323232168126666626488212363636361212

28168648482432323216812666662648821236363636121224

2816864848243232321681266666264882123636363612122432

281686484824323232168126666626488212363636361212243264

28168648482432323216812666662648821236363636121224326416

281686484824323232168126666626488212363636361212243264162

2816864848243232321681266666264882123636363612122432641622

28168648482432323216812666662648821236363636121224326416226

2816864848243232321681266666264882123636363612122432641622618

281686484824323232168126666626488212363636361212243264162261818

28168648482432323216812666662648821236363636121224326416226181818

2816864848243232321681266666264882123636363612122432641622618181818

281686484824323232168126666626488212363636361212243264162261818181818

28168648482432323216812666662648821236363636121224326416226181818181818

2816864848243232321681266666264882123636363612122432641622618181818181818

28168648482432323216812666662648821236363636121224326416226181818181818186

281686484824323232168126666626488212363636361212243264162261818181818181862

2816864848243232321681266666264882123636363612122432641622618181818181818622

28168648482432323216812666662648821236363636121224326416226181818181818186222

281686484824323232168126666626488212363636361212243264162261818181818181862224

2816864848243232321681266666264882123636363612122432641622618181818181818622248

281686484824323232168126666626488212363636361212243264162261818181818181862224812

2816864848243232321681266666264882123636363612122432641622618181818181818622248126

281686484824323232168126666626488212363636361212243264162261818181818181862224812612...

281686484824323232168126666626488212363636361212243264162261818181818181862224812612

244612412688888888888884812448328 <— hit (79 étapes – je crois)

Quant aux nombres-Finex, ils me donnent déjà mal au crâne, zappons.

__________

Update August 23rd, 2021

Scott Shannon wrote to me a couple of hours ago:

> This is really tricky! Its not obvious what ones repeat forever and what ones finally stop. Im currently using an arbitrary cutoff of if it sees more 'x' 8's in a row it gives up. So theres a VERY good chance there may be ones that terminate that have a huge number of 8's at some point that I didnt find. Do you have any insights into how one can detect repeats? [Me: no] If not it's really hard to know how long to let the string grow for before quitting.

It seems numbers 122.... have a long sequence before terminating. Found 122222 at index 21368:

12222224444481616161632866666661861648363636363636684866243224181818181818181818181836483232483612812648488888888888888888888824182432246668322418621682122432323264646464646464646464646464646464646464641684816812648243636482464848481226481622481266666122424242424242424242424242424242424242424242424242424242424242424242424242424244648323286488212243216812181818243216824243232323232824122432861248328212363636366248888888888888888888888888888888888888888888888888888888888888888888888888888816242432246661648243264162248126264882288888168126264816888126666666661616842481261648628322461616226181818181818183612832646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464648612888126482436366624321681261224461248328212121224326416416646464648648821212122432864864648212363636363636363666664832883282126624324812162464824666612412688888888888882418621624612242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424243248621664648212243216812181818363612812626488212624816246283224616162222224812612244624463624242424242424324824326416222222481261648243248242424321622618181818181818181818181818181836363636243224616642461616221236128126832822612824243216824363636628421248646464646464646464646464168484812261282462488888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888881268324812263624242432162248126264882288888241818186216821212122432641622121283286128241216246482466661244444832821262481624128162418181288888888888881268321681261224461244444832821266243216812683216888881262612412688888888888888888888888888888241818181818181812812648246636248824666612422618621682124824616164126216168881262648168121818181818361216328228324824242424242424242424242424242424242424242424244648323232328241262161682412832646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464646464821248246832824121818128888812626124832821212122432641641664646464168488888481226481622222248126122446124222216246164862161684226128242432168243636366281616161632246161622121283286128421686128488882166464646464646464646464648212482462648821262481624628161616163224616162212361281262648821248246264864646464821212126284212486464646464646464646464646464646464646464646464646464646416848888888888888821682122432168243618181283264168243636366288412684812264816228321682466662442121226664864648212121224328648822888888888241862261861616416246832168888888888888888888888888888888888888888888881624243224666666616168421212266648168421624612242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242424242432162283216824482461616842288882166464646482121212628322461616222222481261224462446362424242424242446483232646464643232824122432861244444832821262481624628844426128246624324812266648328412621616888126264816812181818181836121686666666186482466661242222162461648621632826484862163232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