Erasing even digits

We S start with a(1) = k, multiply k by 3 and iterate. When even digits appear in the result, we erase them, concatenate what is left and go on.

S = 1, 3, 9, (2)7, (2)1, loop

S = 2, (6) END

S = 4, 1(2), ... loop

S = 5, 15, (4)5, loop

S = 6, 1(8), ... loop

S = 8, (24) END

S = 10, 3(0), ... loop

S = 11, 33, 99, (2)97, (2)91, (2)73, (2)19, 57, 171, 513, 1539, (46)17, 51, 153, (4)59, 177, 531, 1593, (4)779, (2)337, 1(0)11, 333, 999, (2)997, (2)991, (2)973, (2)919, (2)757, (22)71, (2)13, 39, 117, 351, 1(0)53, loop

S = 12, 3(6), ... loop

S = 14, (42) END

S = 16, (48) END

S = 18, 5(4), ... loop

S = 20, (60) END

S = 21, (6)3, ... loop

S = 22, (66) END

S = 23, (6)9, ... loop

S = 24, 7(2), ... loop

S = 25, 75, (22)5, ... loop

S = 26, 7(8), ... loop

S = 27, (8)1, ... loop

S = 28, (84) END

S = 29, (8)7, ... loop

S = 30, 9(0), ... loop

S = 31, 93, (2)79, (2)37, 111, ... loop

S = 32, 9(6), ... loop

S = 34, 1(02), ... loop

S = 35, 1(0)5, ... loop

S = 36, 1(08), ... loop

S = 37, 111, ... loop

S = 38, 11(4), ... loop

S = 40, 1(20), ... loop

S = 41, 1(2)3, ... loop

S = 42, 1(26), ... loop

S = 43, 1(2)9, ... loop

S = 44, 13(2), ... loop

S = 45, 135, (40)5, ... loop

S = 46, 13(8), ... loop

S = 47, 1(4)1, ... loop

S = 48, 1(44), ... loop

S = 49, 1(4)7, ... loop

S = 50, 15(0), ... loop

Etc.

Questions

a) Will 11 be beaten (in length) at some point?

b) Does every integer have at least one ancestor?

c) If yes, does it mean that a sequence of any length is possible – working backwards?

____________________

Next day update, Feb 1st 2024

Giorgos K. was quick to answer these questions – than came Jean-Marc Falcoz:

> GK

> Hi Eric!

> Here is my view on your questions..

> a) 11 can be beaten. Here are the first numbers that make records for longest chains

#terms -> {terms} -> last term that creates a loop

33->{11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}->153

34->{38,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}->153

39->{259,777,331,993,979,937,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}->153

40->{653,1959,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}->153

41->{3037,9111,7333,1999,5997,17991,53973,11919,35757,1771,5313,15939,717,151,53,159,77,31,93,79,37,111,333,999,997,991,973,919,757,71,13,39,117,351,153,59,177,531,1593,779,337}->111

42->{12379,37137,11111,33333,99999,99997,99991,99973,99919,99757,9971,9913,9739,917,751,53,159,77,31,93,79,37,111,333,999,997,991,973,919,757,71,13,39,117,351,153,59,177,531,1593,779,337}->111

43->{13239,39717,119151,35753,1759,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}->153

46->{66657,199971,599913,1799739,539917,119751,35953,1759,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}->153

54->{1037037,3111111,9333333,7999999,3999997,11999991,35999973,17999919,53999757,1199971,3599913,1799739,539917,119751,35953,1759,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}->153

> As we can see in most cases we meet the number 11 which falls into a loop for 153.

> b) We can find an ancestor by searching backwards. The problem is that if we want to extend the sequence, this ancestor must only have odd digits and this makes things harder...

> c) So, I believe that every length is possible, but finding the right ancestor is difficult.

> Lets try for example to extend our last seq of length 54:

{1037037,3111111,9333333,7999999,3999997,11999991,35999973,17999919,53999757,1199971,3599913,1799739,539917,119751,35953,1759,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}

> The first term (1037037) has zeroes in it. So, we should start from the second term looking for ancestors:

3111111 -> 1137137137 here is the first valid ancestor. we continue:

1137137137 -> 37913791379 We found one more ancestor, but the numbers are getting huge for a computer...

> In this way, we have a new record of 55 terms:

{37913791379,1137137137,3111111,9333333,7999999,3999997,11999991,35999973,17999919,53999757,1199971,3599913,1799739,539917,119751,35953,1759,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}

____________________

> JMF

> 11 n'est qu'un petit joueur comme point de départ de ta suite :o)

> Voici quelques meilleurs départs :

{38,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351} (puis 1053=153=loop)

{259,777,331,993,979,937,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351} (puis...)

{653,1959,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}

{3037,9111,7333,1999,5997,17991,53973,11919,35757,1771,5313,15939,717,151,53,159,77,31,93,79,37,111,333,999,997,991,973,919,757,71,13,39,117,351,153,59,177,531,1593,779,337}

{12379,37137,11111,33333,99999,99997,99991,99973,99919,99757,9971,9913,9739,917,751,53,159,77,31,93,79,37,111,333,999,997,991,973,919,757,71,13,39,117,351,153,59,177,531,1593,779,337}

{13239,39717,119151,35753,1759,577,1731,5193,15579,737,11,33,99,97,91,73,19,57,171,513,1539,17,51,153,59,177,531,1593,779,337,111,333,999,997,991,973,919,757,71,13,39,117,351}

> Et surtout, les nombres de type {11111111111...1} donnent naissance à des suites très longues, par exemple, le nombre : 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

donne naissance à une suite de 940 termes (ci-dessous), mais nul doute qu'avec un point de départ de même type, et encore plus grand, on arrive à bien plus de termes. On arrive même probablement à un nombre arbitrairement grand de termes, en choisissant un point de départ assez grand.

S940 termes=

{11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111,33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333,99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999,99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999997,99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999991,99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999973,99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999919,99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999757,9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999971,99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999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____________________

> The approach of Jean-Marc Falcoz using repunits is very interesting!

I managed to find a formula that tells us how many terms a repunit can produce.

And it is increasing, which means that we can easily compute the length of arbitrary large seqs starting from a given repunit!

Here is the formula (it seems complicated but it is not):

First of all we will call r(d) a repunit with d digits

So, r(3)=111, r(5)=11111 and so on

If we want to know how many terms a repunit r(d) produces we do the following:

The formula works for d>=8.

First step:

We take the quotient (integer part) of (d-8)/11 and multiply it by 20. We will call this A.

So, A = 20 * floor((d-8)/11)

Second step:

We find the remainder (residue) of (d-8)/11 and we add 1. We will call this B;

It is obvious that B will always be 1,2,3,4,5,6,7,8,9,10 or 11

Third step:

We use the key numbers {53, 53, 53, 53, 48, 51, 51, 53, 60, 54, 60}

The first key number is 53, the 2nd 53 the 5th 48... the 11th 60.

Final step:

The number of terms for a repunit r(d) is A + Bth key number

Let's take an example

The repunit that JMF used, has 500 digits

(500-8) / 11 = 492 / 11

The quotient of this division is 44 and the remainder is 8

A = 44 * 20 = 880 and

B is the remainder +1, B=8+1=9 and we look up and see that the 9th key number is 60.

So, A+Bth key number = 880 + 60 = 940 correct!

Now we can know how many terms any huge repunit can produce.

For example a repunit with 1million digits will produce 1818208 terms!

Also a repunit of 10^20 terms would produce 181818181818181818208 terms and so on to infinity!

____________________

ÉA

Magnifique Giorgos and Jean-Marc – and thank you very much for these dizzying numbers!

(Dall-e creation)

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