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:

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-périodicitéavait 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.