Fun with roots



We look for the digital root of a(n) and concatenate it to a(n). Then iterate. What do successive whole numbers produce?

a(1) = 1, 11, 112, 1124, 11248, 112487, 1124875, 11248751, 112487512, 1124875124, 11248751248, 112487512487, 1124875124875, ...
The yellow pattern is repeated ad infinitum.

a(1) = 2, 22, 224, 2248, 22487, 224875, 2248751, ...
The cyan pattern will also be repeated ad infinitum.

a(1) = 3, 33, 336, 3363, 33636, 336363, 3363636, 33636363, 336363636, ...
The green pattern will be repeated for ever.

a(1) = 4, 44, 448, 4487, 44875, 448751, 4487512, ... The grey pattern, etc.
a(1) = 5, 55, 551, 5512, 55124, 551248, 5512487, 5512487, ... 
a(1) = 6, 66, 663, 6636, 66363, ...
a(1) = 7, 77, 775, 7751, 77512, 775124, 7751248, 7751248, ...
a(1) = 8, 88, 887, 8875, 88751, 887512, 8875124, ...
a(1) = 9, 99, 999, 9999, 99999, ...
a(1) = 10, 101, 1012, 10124, 101248, 1012487, 10124875, ...
a(1) = 11, 112, 1124, 11248, 112487, 1124875, ...
a(1) = 12, 123, 1236, ...
a(1) = 13, 134, 1348, 13487, 137875, 1348751, 13487512, ...
a(1) = 14, 145, 1451, 14512, 145124, 1451248, 14512487, ...
a(1) = 15, 156, 1563, ...
a(1) = 16, 167, 1675, 16751, 167512, 167124, 16751248, ...
a(1) = 17, 178, 1787, 17875, 178751, 1787512, 17875124, ...
a(1) = 18, 189, 1899, ...
a(1) = 19, 191, 1912, 19124, 191248, 1912487, 19124875, ...
a(1) = 20, 202, 2024, 20248, 202487, 2024875, 20248751, ...
etc.

Question #1
What is the smallest integer a(1) that will produce a number that contains at least one copy of all the digits from 0 to 9? 
The answer might be the number written (in white “invisible” ink) here:10369
Question #2
How does the sequence Z behave if…
1) we start Z with a(1) = 1;
2) we pause Z immediately after a prime number appears or immediately after a number with digital root 3, 6 or 9 appears;
3) we then extend Z with the smallest integer not yet present in Z;
4) and we resume the iteration.

Here is the author's try:
Starting with a(1) = 1 we have (primes in red)
S = 1, 11 – we pause as 11 is prime. We extend S with 2:
S = 1, 11, 2 – we pause as 2 is prime. We extend S with 3:
S = 1, 11, 2, 3 – we pause as 3 is prime (and has a DR of 3). We extend S with 4:
S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487 – we pause as this is a prime. We extend S with 5:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5 – we pause as 5 is prime. We extend S with 6:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6 – we pause as 6 has a DR of 6. We extend S with 7:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7 – we pause as 7 is prime. We extend S with 8:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887 – which is prime. We extend S with 9:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9 – we pause as 9 has a DR of 9. We extend S with 10:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101 – which is prime. We extend S with 12 (as 11 is already present in S):

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12 – we pause as 12 has a DR of 3. We extend S with 13:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13 – which is prime. We pause and extend S with 14:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451 – which is prime. We extend S with 15:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451, 15 – which has a DR of 6. We extend S with 16:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451, 15, 16, 167 – which is prime. We extend S with 17:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451, 15, 16, 167, 17 – which is prime. We extend S with 18:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451, 15, 16, 167, 17, 18 – which has a DR of 6. We extend S with 19:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451, 15, 16, 167, 17, 18, 19 – which is prime. We extend S with 20:

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451, 15, 16, 167, 17, 18, 19, 20, 202, 2024, … which is now! (and time to let the reader explore Z by himself!)

P.-S.
The next prime number of S might be 20248751248751248751248751248751248751248751248751 [a 50-digit term repeating 8 times the yellow pattern we know! Will Z host a longer prime soon?]
;-D
____________________
First of August update
Maximilian Hasler was quick to send this (to Math Fun):
MH
> The "very next" prime in that sequence is indeed much larger, it's Z[280] =
22487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487
with 191 digits.

Hans Havermann was quick to post this (to Math Fun):
HH
> Looking only at record large primes, after z(280) we find z(1064) with 766 digits. The one after this is a doozie: z(3737+d) with d decimal digits, d > 32000.

Hans has wonderful blog where he mentions the seq:
The b-file of the above S seq is there:
... the size of the primes involved is absolutely ...
Here are the first 100 terms of S (they have been submitted to the OEIS, here):

S = 1, 11, 2, 3, 4, 44, 448, 4487, 44875, 448751, 4487512, 44875124, 448751248, 4487512487, 44875124875, 448751248751, 4487512487512, 44875124875124, 448751248751248, 4487512487512487, 5, 6, 7, 8, 88, 887, 9, 10, 101, 12, 13, 14, 145, 1451, 15, 16, 167, 17, 18, 19, 20, 202, 2024, 20248, 202487, 2024875, 20248751, 202487512, 2024875124, 20248751248, 202487512487, 2024875124875, 20248751248751, 202487512487512, 2024875124875124, 20248751248751248, 202487512487512487, 2024875124875124875, 20248751248751248751, 202487512487512487512, 2024875124875124875124, 20248751248751248751248, 202487512487512487512487, 2024875124875124875124875, 20248751248751248751248751, 202487512487512487512487512, 2024875124875124875124875124, 20248751248751248751248751248, 202487512487512487512487512487, 2024875124875124875124875124875, 20248751248751248751248751248751, 202487512487512487512487512487512, 2024875124875124875124875124875124, 20248751248751248751248751248751248, 202487512487512487512487512487512487, 2024875124875124875124875124875124875, 20248751248751248751248751248751248751, 202487512487512487512487512487512487512, 2024875124875124875124875124875124875124, 20248751248751248751248751248751248751248, 202487512487512487512487512487512487512487, 2024875124875124875124875124875124875124875, 20248751248751248751248751248751248751248751, 202487512487512487512487512487512487512487512, 2024875124875124875124875124875124875124875124, 20248751248751248751248751248751248751248751248, 202487512487512487512487512487512487512487512487, 2024875124875124875124875124875124875124875124875, 20248751248751248751248751248751248751248751248751, 21, 22, 224, 2248, 22487, 224875, 2248751, 22487512, 224875124, 2248751248, 22487512487,...

Merci beaucoup, Maximilian et Hans !

The author's conclusion was sent to Math Fun:
ÉA
> Marvelous vertigos – Hitchcock is really a small player when compared to the Math Fun team -- thanks !-)






Commentaires

  1. MFH30 juillet 2024 à 01:04

    Ce commentaire a été supprimé par l'auteur.

    RépondreSupprimer
  2. MFH30 juillet 2024 à 01:17

    Next prime is Z[280] = 22487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487512487 with 191 digits.

    RépondreSupprimer

Enregistrer un commentaire

Posts les plus consultés de ce blog

A square for three (chess)

Image
(English translation after the French text) Voici cinq problèmes d’échecs disjoints : a ) combien faut-il de coups au minimum pour que trois pions soient capturés sur la même case ? b ) trois tours c ) trois cavaliers d ) trois fous e ) trois dames Here are five disjointed chess problems: a) how many moves are required at least for three pawns to be captured on the same square? b) three rooks c) three knights d) three bishops e) three queens Solutions a ) (corrigé grâce à Orion , ici ) 3,5 coups pour 3 pions capturés en b5 (4 moves for White and 3 moves for Black   — 3 pawns  captured on b5)  b ) 5  coups pour 3 tours capturées en a3 (ci-dessous – merci aussi à Orion !-) (5 moves for White and 5 moves for Black  — 3 rooks  captured on a3  hereunder )  c ) 5  coups pour 3 cavaliers capturés en d5  ( ci-dessous) (5 moves for White and 5 moves for Black   — 3 knights  captured on d5  hereunder )  (merci Orion !-) d ) ...
Lire la suite

Le tripalin se présente

Image
Un tripalin est constitué de trois images. Chaque image illustre un substantif. Accolés, ces trois substantifs forment une chaîne palindromique. Laquelle nous vous invitons à trouver.  Exemple ci-dessous. Le premier mot est COL-BLEU Le second DUEL Le troisième BLOC . Ces mots forment la chaîne COLBLEUDUELBLOC qui est bien palindromique. Je posterai chaque jeudi une nouvelle salve de tripalins . Rien que des noms illustrés – pas d’adjectifs, pas de verbes conjugués, pas d’adverbes (parfois un prénom ou un nom propre – mais seulement quand il faut). À la demande générale (de mon neveu), chaque tripalin sera surmonté de son numéro d ’ ordre ( octothorpe noir) et de trois nombres bleus. Ces nombres reprennent les trois longueurs (en lettres) des trois mots présents dans le tripalin. On a par exemple ci-dessous [ #0 ↓ 7+4+4 ] car COL-BLEU comporte  7  lettres, DUEL   4  et BLOC   4 . Le numéro d ’ ordre est zéro. #0 ↓ 7+4+4 Voilà – tous retours critiques, con...
Lire la suite

Some strings au cinéma Galeries

  Lettre ouverte au cinéma Galeries   Bonsoir à tous,   Je viens de voir pour la seconde fois chez vous le beau film de Léos Carax (la première fois c’était le 26 juin en présence du réalisateur, au BRIFF). Apparut à l’écran aujourd’hui, avant la projection proprement dite, un avis que je n’avais pas remarqué en juin, expliquant un projet baptisé Some strings (« Quelques ficelles »), visible au sous-sol. Cet avis semblait faire l’éloge d’un poète palestinien mort à Gaza, lors d’un bombardement. Je n’ai pas bien compris de quoi il s’agissait, mais les mots utilisés m’ont interpellé. Après le Carax  je suis descendu voir l’exposition Some strings . J’en ai lu tous les cartels et avis placardés dans votre cinéma – et j’aimerais vous poser, si vous le voulez bien, quelques questions (je suis journaliste, vous m’avez d’ailleurs gentiment donné un « Exo » pour le Carax ).   Cette phrase, par exemple, omniprésente au cinéma Galeries, mériterait de...
Lire la suite