Waterfalls

(of multiplications)

 

 

Hello SeqFans,

Start S with a(1) > 9

Now a(n) is simply the result of a multiplication : [the (n-1)th digit of S] x [the n-th digit of S]

 

Let’s start S respectively with a(1) = [10, 11, 12, 13, 14, 15, 16, ...]

 

S = 10,0,0,0,0,0,0,0,...

 

S = 11,1,1,1,1,1,1,1,...

                                                    __

S = 12,2,4,8,32,24,6,4,8,24,24,32,16,8,8,8,12,6,2,6,48,64,64,8,2,12,12,12,24,32,...

                                                   __

S = 13,3,9,27,18,14,7,8,8,4,28,56,64,32,8,16,40,30,36,24,12,6,16,8,6,24,0,0,0,0,18,...

                                                          __

S = 14,4,16,4,6,24,12,8,4,2,16,32,8,2,6,18,6,16,16,12,6,8,48,6,6,6,6,6,2,12,48,32,32,...

 

S = 15,5,25,10,10,5,0,0,0,0,0,0,0,0,...

                          __

S = 16,6,36,18,18,6,8,8,8,48,48,64,64,32,...

                                        __

(the "overlined" multiplications --like 48-- produce the last computed so far term of some of the above sequences)

 

Question:

What are the integers >9 which end on a fixed point (a fixed point like 0 or 1) or end in a loop?

Respectively, what are the integers escaping fixed points and loops?

 

Though a 2-digit multiplication cannot produce a term > 81, I’ve noticed that 13, for instance, never ends in a loop -- because the first "0,0,0,0" pattern will turn later in a "0,0,0,0,0" (5-zeroes) pattern, which will turn later in a 6-zeroes pattern, than 7-zeroes, etc. -- and between those 0-patterns you will always count more and more digits...

 

Best,

É.

 

__________

 

Hans Havermann was quick to compute and draw this:

 

> For each n = 10, 11, ... 200, first a graph of 2000 terms of S, followed immediately by a graph of 2000 digits of S:

> http://chesswanks.com/num/waterfalls/

 

 

Maximilian Hasler has sent me this comment (in French) and the S-sequences for n= 12 to 20 (illustrated below with the diagrams provided by Hans Havermann’s link):

 

(...) il est probable que tout nombre finit par entrer dans un cycle "quasi-périodique" qui peut être complètement décrit (du genre:  f(n) fois ça, g(n) fois ça, h(n) fois ça... et on recommence avec n+1 (...)

 

[Programme]:

casc = (a,n=200)->my(S=eval(Vec(Str(a))));for(i=1,n,i>1&S=concat(vecextract(S,"^1"),eval(Vec(Str(a=S[1]*S[2]))));print1(a","))

 

gp > casc(12,400)

12, 2, 4, 8, 32, 24, 6, 4, 8, 24, 24, 32, 16, 8, 8, 8, 12, 6, 2, 6, 48, 64, 64, 8, 2, 12, 12, 12, 24, 32, 48, 24, 24, 24, 32, 16, 2, 2, 2, 2, 2, 2, 4, 8, 12, 6, 8, 32, 16, 8, 8, 8, 8, 8, 12, 6, 2, 6, 12, 4, 4, 4, 4, 4, 8, 32, 8, 2, 12, 48, 24, 6, 2, 6, 48, 64, 64, 64, 64, 8, 2, 12, 12, 12, 6, 2, 8, 16, 16, 16, 16, 32, 24, 6, 16, 16, 2, 2, 8, 32, 16, 8, 24, 12, 12, 24, 32, 48, 24, 24, 24, 24, 24, 24, 24, 32, 16, 2, 2, 2, 2, 2, 2, 12, 12, 16, 8, 6, 6, 6, 6, 6, 6, 6, 18, 6, 4, 8, 24, 6, 6, 6, 6, 12, 4, 16, 24, 6, 2, 6, 48, 16, 8, 4, 2, 2, 2, 4, 8, 12, 6, 8, 32, 16, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 12, 6, 2, 6, 12, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 6, 48, 48, 36, 36, 36, 36, 36, 36, 6, 8, 48, 24, 32, 16, 8, 24, 36, 36, 36, 6, 2, 8, 4, 6, 12, 8, 24, 12, 12, 24, 32, 8, 6, 48, 32, 8, 4, 4, 8, 32, 8, 2, 12, 48, 24, 6, 2, 6, 48, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 8, 2, 12, 12, 12, 6, 2, 8, 16, 16, 16, 16, 8, 4, 4, 4, 4, 12, 24, 32, 32, 32, 24, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 36, 48, 32, 32, 16, 8, 12, 6, 2, 6, 48, 16, 8, 12, 18, 18, 18, 18, 18, 36, 12, 16, 32, 24, 6, 2, 16, 16, 8, 4, 2, 2, 2, 4, 8, 12, 6, 16, 48, 24, 32, 24, 6, 16, 32, 16, 32, 24, 6, 16, 16, 2, 2, 8, 32, 16, 8, 24, 12, 12, 24, 32, 48, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 32, 16, 2, 2, 2, 2, 2, 2, 12, 12, 16, 8, 6, 6, 6, 6, 6, 6, 6, 48, 32, 16, 16, 16, 4, 2, ...

 

 

gp > casc(13,400)

13, 3, 9, 27, 18, 14, 7, 8, 8, 4, 28, 56, 64, 32, 8, 16, 40, 30, 36, 24, 12, 6, 16, 8, 6, 24, 0, 0, 0, 0, 18, 12, 8, 4, 2, 12, 6, 6, 48, 48, 12, 8, 0, 0, 0, 0, 0, 8, 8, 2, 16, 32, 8, 2, 2, 12, 36, 24, 32, 32, 32, 8, 2, 16, 0, 0, 0, 0, 0, 0, 64, 16, 2, 6, 18, 6, 16, 16, 4, 2, 2, 6, 18, 12, 8, 12, 6, 6, 6, 6, 6, 16, 16, 2, 6, 0, 0, 0, 0, 0, 0, 0, 24, 4, 6, 12, 12, 6, 8, 48, 6, 6, 6, 6, 24, 8, 4, 12, 6, 8, 8, 2, 16, 8, 2, 12, 36, 36, 36, 36, 6, 6, 6, 6, 12, 12, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 24, 6, 2, 2, 2, 12, 48, 32, 32, 48, 36, 36, 36, 12, 8, 32, 32, 4, 2, 12, 48, 64, 16, 2, 6, 48, 16, 2, 2, 6, 18, 18, 18, 18, 18, 18, 18, 36, 36, 36, 36, 6, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 6, 12, 8, 24, 12, 4, 4, 2, 2, 8, 32, 24, 6, 6, 6, 8, 32, 24, 18, 18, 18, 18, 18, 6, 2, 16, 24, 6, 6, 6, 8, 8, 2, 2, 8, 32, 48, 24, 4, 6, 12, 12, 24, 32, 8, 6, 12, 4, 12, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 24, 18, 18, 18, 18, 18, 18, 18, 36, 12, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 6, 2, 16, 16, 8, 4, 2, 8, 16, 8, 4, 16, 24, 6, 4, 8, 24, 36, 36, 48, 24, 6, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 48, 12, 2, 6, 12, 8, 24, 36, 36, 48, 64, 16, 4, 16, 24, 6, 8, 32, 16, 8, 16, 24, 6, 2, 2, 2, 4, 8, 12, 6, 16, 48, 6, 2, 8, 4, 2, 12, 48, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 16, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 24, 18, 6, 2, 8, 16, 0, 0, 0, 0, 0, 0, ...

 

 

gp > casc(14,400)

14, 4, 16, 4, 6, 24, 24, 12, 8, 8, 8, 4, 2, 16, 64, 64, 32, 8, 2, 6, 36, 24, 24, 24, 12, 6, 16, 16, 12, 18, 18, 12, 8, 8, 8, 8, 8, 4, 2, 12, 6, 6, 6, 6, 6, 2, 2, 8, 8, 8, 8, 2, 16, 64, 64, 64, 64, 32, 8, 2, 2, 12, 36, 36, 36, 36, 12, 4, 16, 64, 64, 64, 16, 2, 6, 36, 24, 24, 24, 24, 24, 24, 24, 12, 6, 16, 16, 4, 2, 2, 6, 18, 18, 18, 18, 18, 18, 18, 6, 2, 8, 4, 6, 36, 24, 24, 24, 24, 24, 4, 6, 12, 12, 18, 18, 12, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 2, 12, 6, 6, 6, 6, 24, 8, 4, 12, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 48, 12, 16, 32, 24, 18, 18, 12, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 24, 6, 2, 2, 2, 2, 8, 8, 8, 8, 2, 16, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 8, 2, 2, 12, 36, 36, 36, 12, 8, 32, 32, 4, 2, 12, 48, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 8, 2, 2, 6, 18, 6, 4, 8, 4, 8, 8, 8, 8, 2, 16, 64, 64, 64, 64, 64, 64, 64, 64, 8, 6, 12, 8, 24, 12, 4, 4, 4, 16, 64, 64, 64, 16, 2, 6, 36, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 12, 6, 16, 16, 4, 2, 2, 6, 18, 18, 18, 18, 18, 6, 2, 16, 24, 6, 6, 6, 8, 8, 2, 2, 8, 32, 48, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 12, 6, 6, 6, 16, 16, 4, 12, 6, 8, 48, 24, 32, 32, 32, 64, 64, 64, 16, 2, 6, 36, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 32, 48, 6, 2, 16, 16, 8, 4, 2, 8, 16, 16, 4, 6, 36, 24, 24, 24, 24, 24, 4, 6, 12, 12, ...

 

 

gp > casc(15,400)

15,5,25,10,10,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, ...

 

(03:41) gp > casc(16,400)

16, 6, 36, 18, 18, 6, 8, 8, 8, 48, 48, 64, 64, 32, 32, 32, 32, 48, 24, 24, 24, 12, 6, 6, 6, 6, 6, 6, 6, 8, 32, 16, 8, 8, 8, 8, 8, 4, 2, 12, 36, 36, 36, 36, 36, 36, 48, 24, 6, 2, 6, 48, 64, 64, 64, 64, 32, 8, 2, 2, 6, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 24, 32, 16, 8, 24, 12, 12, 24, 32, 48, 24, 24, 24, 24, 24, 24, 24, 12, 6, 16, 16, 4, 12, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 8, 12, 6, 2, 6, 48, 16, 8, 4, 2, 2, 2, 4, 8, 12, 6, 8, 32, 16, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 2, 12, 6, 6, 6, 6, 24, 4, 2, 12, 48, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 8, 6, 48, 8, 2, 12, 12, 12, 24, 32, 8, 6, 48, 32, 8, 4, 4, 8, 32, 8, 2, 12, 48, 24, 6, 2, 6, 48, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 8, 2, 2, 12, 36, 36, 36, 12, 8, 16, 8, 2, 2, 8, 32, 48, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 32, 48, 24, 32, 64, 16, 2, 2, 2, 2, 2, 2, 4, 8, 12, 6, 16, 48, 24, 32, 24, 6, 16, 32, 16, 32, 24, 6, 16, 16, 2, 2, 8, 32, 16, 8, 24, 12, 12, 24, 32, 48, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 12, 6, 16, 16, 4, 2, 2, 6, 18, 18, 18, 18, 18, 6, 2, 16, 8, 6, 48, 16, 4, 16, 24, 6, 8, 32, 16, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...

 

 

gp > casc(17,400)

17, 7, 49, 28, 36, 18, 16, 24, 18, 6, 8, 8, 6, 12, 8, 4, 8, 48, 48, 64, 48, 6, 2, 16, 32, 32, 32, 32, 32, 32, 48, 24, 16, 32, 48, 12, 2, 6, 18, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 32, 16, 8, 4, 6, 18, 6, 8, 32, 8, 2, 4, 12, 6, 8, 48, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 48, 24, 6, 2, 6, 48, 32, 24, 6, 8, 48, 48, 24, 6, 16, 16, 8, 4, 2, 12, 48, 32, 32, 24, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 24, 32, 16, 8, 24, 12, 12, 24, 32, 24, 6, 4, 8, 24, 48, 32, 32, 32, 32, 16, 8, 24, 6, 6, 6, 6, 48, 32, 8, 2, 2, 8, 32, 24, 6, 6, 6, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 8, 12, 6, 2, 6, 48, 16, 8, 4, 2, 2, 2, 4, 8, 12, 6, 4, 8, 24, 24, 32, 16, 8, 16, 32, 24, 6, 6, 6, 6, 6, 6, 6, 2, 6, 48, 16, 8, 24, 36, 36, 36, 24, 32, 24, 6, 16, 16, 4, 16, 24, 6, 4, 8, 24, 36, 36, 24, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 8, 6, 48, 8, 2, 12, 12, 12, 24, 32, 8, 6, 48, 32, 8, 4, 4, 8, 32, 8, 2, 12, 24, 32, 16, 8, 8, 8, 12, 6, 2, 6, 48, 8, 6, 18, 6, 4, 8, 24, 36, 36, 36, 36, 36, 36, 12, 12, 24, 32, 8, 6, 48, 16, 8, 12, 18, 18, 18, 18, 18, 12, 8, 12, 6, 4, 8, 24, 6, 6, 6, 6, 24, 4, 6, 12, 8, 24, 24, 32, 16, 8, 12, 18, 18, 18, 12, 8, 12, 6, 6, 6, 6, 6, 12, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, ...

 

 

gp > casc(18,400)

18, 8, 64, 48, 24, 16, 32, 16, 8, 4, 6, 18, 6, 2, 6, 48, 32, 24, 6, 8, 48, 12, 12, 24, 32, 24, 6, 4, 8, 24, 48, 32, 32, 8, 2, 2, 2, 4, 8, 12, 6, 4, 8, 24, 24, 32, 16, 8, 16, 32, 24, 6, 6, 6, 16, 16, 4, 4, 8, 32, 8, 2, 12, 24, 32, 16, 8, 8, 8, 12, 6, 2, 6, 48, 8, 6, 18, 6, 4, 8, 24, 36, 36, 6, 6, 6, 6, 24, 16, 32, 24, 6, 16, 16, 2, 2, 4, 8, 12, 6, 2, 6, 48, 64, 64, 8, 2, 12, 12, 12, 24, 32, 64, 48, 6, 8, 48, 24, 32, 16, 8, 12, 18, 18, 18, 36, 36, 36, 36, 12, 8, 4, 6, 18, 6, 4, 8, 24, 6, 6, 6, 6, 12, 4, 8, 32, 8, 2, 12, 12, 12, 24, 32, 48, 24, 24, 24, 32, 16, 2, 2, 2, 2, 2, 2, 4, 8, 12, 6, 12, 24, 16, 32, 48, 48, 32, 32, 16, 8, 12, 6, 2, 6, 48, 8, 2, 2, 8, 8, 8, 8, 8, 24, 18, 18, 18, 18, 18, 18, 18, 6, 2, 16, 32, 24, 6, 8, 48, 24, 32, 16, 8, 24, 36, 36, 36, 6, 2, 8, 32, 24, 6, 16, 16, 2, 2, 2, 2, 2, 2, 4, 8, 12, 6, 8, 32, 16, 8, 8, 8, 8, 8, 12, 6, 2, 6, 12, 4, 4, 4, 4, 4, 8, 32, 8, 2, 12, 6, 2, 4, 8, 4, 6, 18, 6, 8, 32, 32, 32, 24, 6, 6, 6, 2, 6, 48, 8, 2, 12, 12, 12, 24, 32, 64, 16, 4, 16, 64, 64, 64, 64, 16, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 48, 12, 2, 6, 18, 6, 4, 8, 24, 48, 32, 32, 16, 8, 12, 6, 2, 6, 48, 16, 8, 12, 18, 18, 18, 18, 18, 36, 12, 16, 24, 6, 4, 8, 24, 6, 6, 6, 6, 12, 4, 4, 4, 4, 4, 8, 32, 8, 2, 12, 48, 24, 6, 2, 6, 48, 64, 64, 64, 64, 8, 2, 12, 12, 12, 6, 2, 8, 16, 16, 16, 16, 32, 24, 6, 16, 16, 2, 2, 12, 12, 8, 32, 32, 24, 6, 8, 48, 48, 24, 6, 6, 6, ...

 

 

gp > casc(19,400)

19, 9, 81, 72, 8, 7, 14, 16, 56, 7, 4, 4, 6, 30, 30, 42, 28, 16, 24, 18, 0, 0, 0, 0, 8, 4, 16, 8, 6, 12, 8, 4, 8, 0, 0, 0, 0, 0, 32, 4, 6, 48, 48, 6, 2, 16, 32, 32, 0, 0, 0, 0, 0, 0, 6, 8, 24, 24, 32, 32, 32, 48, 12, 2, 6, 18, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 48, 16, 8, 8, 8, 12, 6, 6, 6, 6, 6, 8, 32, 8, 2, 4, 12, 6, 8, 48, 36, 36, 0, 0, 0, 0, 0, 0, 0, 0, 32, 8, 6, 48, 64, 64, 8, 2, 12, 36, 36, 36, 36, 48, 24, 6, 16, 16, 8, 4, 2, 12, 48, 32, 32, 24, 18, 18, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 16, 48, 24, 32, 48, 24, 24, 24, 32, 16, 2, 2, 6, 18, 18, 18, 18, 18, 18, 18, 24, 32, 16, 8, 24, 6, 6, 6, 6, 48, 32, 8, 2, 2, 8, 32, 24, 6, 6, 6, 4, 8, 4, 8, 8, 8, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 24, 32, 16, 8, 12, 6, 8, 32, 16, 8, 8, 8, 8, 8, 12, 6, 2, 6, 12, 4, 12, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 8, 12, 6, 2, 6, 48, 16, 8, 24, 36, 36, 36, 24, 32, 24, 6, 16, 16, 4, 16, 24, 6, 4, 8, 24, 36, 36, 24, 32, 32, 32, 64, 64, 64, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 12, 8, 12, 6, 2, 6, 48, 8, 2, 12, 48, 24, 6, 2, 6, 48, 64, 64, 64, 64, 8, 2, 12, 12, 12, 6, 2, 8, 4, 2, 12, 48, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 8, 6, 48, 8, 2, 12, 12, 12, 24, 32, 8, 6, 48, 16, 8, 12, 18, 18, 18, 18, 18, 12, 8, 12, 6, 4, 8, 24, 6, 6, 6, 6, 24, 4, 6, 12, 8, 24, 24, 32, 16, 8, 12, 18, 18, 18, 12, 8, 12, 6, 6, 6, 6, 6, 12, 24, 24, 24, 24, 24, 24, 24, 0, 0, 0, 0, 0, 0, ...

 

 

Thanks to everyone,

Best,

É.

 

__________

 

Update, March 27th, 2012 (in French)

 

Maximilian Hasler:

 

Liste des nombres <1000 qui terminent en [0,0,0,0,0,0...] (liste calculée en approche expérimentale, n’est donc pas 100% rigoureuse, mais semble juste) :

 

10, 15, 20, 25, 30, 35, 40, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 65, 69, 70, 75, 78, 80, 85, 87, 90, 95, 96, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 115, 120, 125, 130, 135, 140, 145, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 165, 169, 170, 175, 178, 180, 185, 187, 190, 196, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 215, 220, 225, 230, 235, 240, 245, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 265, 270, 275, 280, 285, 290, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 315, 320, 325, 330, 335, 340, 345, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 365, 370, 375, 380, 385, 390, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 415, 420, 425, 430, 435, 440, 445, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 465, 470, 480, 485, 490, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 535, 540, 541, 542, 543, 544, 545, 546, 548, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 567, 569, 570, 575, 580, 581, 582, 583, 584, 585, 587, 590, 595, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 615, 619, 620, 625, 630, 635, 640, 645, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 670, 680, 690, 695, 696, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 715, 718, 720, 725, 730, 735, 740, 750, 751, 752, 753, 754, 755, 756, 757, 758, 760, 765, 770, 780, 785, 787, 790, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 815, 817, 820, 825, 830, 835, 840, 845, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 865, 870, 872, 873, 875, 878, 880, 885, 890, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 915, 916, 920, 925, 930, 940, 950, 951, 952, 953, 954, 955, 956, 958, 960, 961, 962, 965, 969, 970, 980, 985, 990, 995, ...

 

Liste des nombres <10^4 ne comportant ni chiffre 0, ni 5 mais qui finissent en [0,0,0,0,0,0,0, ...] :

 

69, 78, 87, 96, 98, 169, 178, 187, 196, 619, 696, 718, 787, 817, 872, 873, 878, 916, 961, 962, 969, 1169, 1178, 1691, 1781, 2987, 6911, 6916, 6961, 6962, 6969, 7817, 7872, 7873, 7878, 8117, 9116, 9696, 9878, ...

 

Liste des nombres < 2000 (après ils sont trop nombreux) qui *ont* un chiffre 0 ou 5 mais ne finissent *pas* en [0,0,0,0,0,0, ...] :

 

59, 195, 295, 395, 475, 495, 519, 533, 534, 536, 537, 538, 539, 547,549, 566, 568, 572, 573, 574, 576, 577, 578, 579, 586, 588, 589, 591,592, 593, 594, 596, 597, 598, 599, 665, 675, 685, 745, 759, 775, 795,895, 935, 945, 957, 959, 975, 1175, 1195, 1245, 1275, 1285, 1295,1325, 1335, 1353, 1356, 1358, 1359, 1365, 1375, 1385, 1395, 1445,1457, 1459, 1465, 1475, 1485, 1495, 1519, 1529, 1539, 1547, 1549,1566, 1567, 1568, 1574, 1577, 1579, 1589, 1593, 1594, 1597, 1615,1635, 1645, 1653, 1654, 1656, 1657, 1658, 1660, 1665, 1675, 1685,1715, 1725, 1735, 1745, 1751, 1752, 1753, 1754, 1756, 1757, 1759,1760, 1765, 1770, 1775, 1790, 1795, 1825, 1835, 1845, 1852, 1854,1856, 1857, 1858, 1859, 1860, 1865, 1880, 1885, 1895, 1925, 1930,1935, 1940, 1945, 1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957,1958, 1959, 1975, 1980, 1985, 1995,

 

Merci Maximilian!

 

À propos du 1660 surligné ci-dessus, Hans Havermann écrit ceci:

 

I had a detailed look at 1660 because I had mis-guessed that a number  containing a zero would end in all-zeros. While not a "loop", the evolution looked regular enough that I wondered if I could describe it. And I believe that I can: After the initial 1660,6,36,0,0,18,18, we get "expanding cycles", for n=1...infinity, of 3n zeros,

                                                                      (4^n+8)/6+1 eights,

                                                                             3n+1 zeros,

                                                                        (4^n+8)/6 sixty-fours,

                                                                             3n+2 zeros,

                                                                      (4^n+8)/3-1 twenty-fours:

 

1660, 6, 36, 0, 0, 18, 18, 0, 0, 0, 8, 8, 8, 0, 0, 0, 0, 64, 64, 0, 0, 0, 0, 0, 24, 24, 24, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 0, 0, 0, 0, 0, 0, 0, 64, 64, 64, 64, 0, 0, 0, 0, 0, 0, 0, 0, 24, 24, 24, 24, 24, 24, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 0, ...

 

n

 zeros

eights

zeros

sixty-fours

zeros

twenty-fours

 

 

 

 

 

 

 

1

 3

 3

 4

 2

 5

 3

2

 6

 5

 7

 4

 8

 7

3

 9

 13

 10

 12

 11

 23

4

 12

 45

 13

 44

 14

 87

5

 15

 173

 16

 172

 17

 343

6

 18

 685

 19

 684

 20

 1367

7

 21

 2733

 22

 2732

 23

 5463

8

 24

 10925

 25

 10924

 26

 21847

9

 27

 43693

 28

 43692

 29

 87383

10

 30

 174765

 31

 174764

 32

 349527

11

 33

 699053

 34

 699052

 35

 1398103

12

 36

 2796205

 37

 2796204

 38

 5592407

13

 39

 11184813

 40

 11184812

 41

 22369623

14

 42

 44739245

 43

 44739244

 44

 89478487

15

 45

 178956973

 46

 178956972

 47

 357913943

16

 48

 715827885

 49

 715827884

 50

 1431655767

17

 51

 2863311533

 52

 2863311532

 53

 5726623063

18

 54

 11453246125

 55

 11453246124

 56

 22906492247

19

 57

 45812984493

 58

 45812984492

 59

 91625968983

20

 60

 183251937965

 61

 183251937964

 62

 366503875927

 

Here’s how it looks:

 

 

 

Maximilian Hasler avait exprimé la même idée de “périodicité paramétrée”:

 

(...) il est probable que tout nombre finit par entrer dans un cycle "quasi-périodique" qui peut être complètement décrit, du genre : f(n) fois ça, g(n) fois ça, h(n) fois ça... et on recommence avec (n+1).

 

 

Cette “quasi-périodicité” avait inspiré à Gilles Esposito-Farèse le commentaire suivant :

 

(...) Je crois qu’il s’agit d’orbites quasi-périodiques comme celles intermédiaires entre les tores périodiques et le chaos total, dans les systèmes hamiltoniens complexes, cf. http://goo.gl/SyXIz. On a généralement (i) des points fixes, (ii) des tores invariants autour de ces points fixes, (iii) des régions quasi-périodiques, (iv) un mélange complet des lignes d’univers.

 

Merci à tous,

à+

É.