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 ) 7  coups pour 3 fous capturés en h6  ( ci-dessous) (7 moves for Whit
Lire la suite

Beautés ?

Image
Beau comme un clavier ? Un deuxième ? (celui de Modotti)   Beau comme le dos d ’ un timbre proposé par Delcampe ? Beau comme une série ukrainienne où les avions râpent le Kremlin ? Belle comme Chargois ? ... belle comme Germaine Richier ? Bel atelier ? Beau comme une pièce de 1 santim ? ... ou comme une pièce de 50  santeem ? ... Byrrh , beau comme un anus ? Beau comme la collection de Marin Karmitz à Beaubourg ? Belles Ménines ? Beau comme Marcel Duchamp ? Beau comme Jean-Luc Godard ? Beau comme la réouverture de l ’ Einsteinturm à Berlin ? Beau comme un anneau dudit ? Beau comme Eisenstein ? Beau comme les fleurs de Pascale hier matin ? Beau comme on dirait la planète Mars ? Beau comme des recherches personnelles ? Beau comme une marelle (vers le ciel) ? Beau comme un triple selfie (au Man Ray ) ? Beau comme le ciel de Sacha , mardi, en face du restaurant La Montagne ? (qui a rouvert) ... beau comme ce bleu-là de Lo ? Beau comme le froid ? Beau comme une Vespa ? Beau comme Cloon
Lire la suite

Underline, reproduce

Image
Dall.e creation If we underline the... « first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh,… » letter of this sentence S 1 , we see that the successive underlined letters simply reproduce S 1 . No big deal. If we decide not to start S 1   with "first" but with "sixth", we can reproduce the new S 6  quite easily: S 6 = sixth , second, third, fourth, fifth, first , seventh, eighth, ninth, tenth, eleventh,… S 6  was reproduced in a slightly less mechanical manner than  S 1 :  just two words of  S 1   swapped here ("first" and "sixth", in red, above). If we start S 8  with "eighth" we just swap again a pair of words of S 1  ("first" and "eighth", in red, hereunder): S 8  = eighth , second, third, fourth, fifth, sixth, seventh, first , ninth, tenth, eleventh,… What about starting S 9  with "ninth"? How many swaps do we need to reproduce S 9 ?  This is more tricky! (Indeed, we
Lire la suite