# [seqfan] Re: Brute force density: triples and cubes

M. F. Hasler oeis at hasler.fr
Thu Oct 2 06:53:33 CEST 2014

```Eric,
the digit's frequency is not uniform at all.
I confirm Bob's observation about something like a p-adic convergence
(i.e., from right to left) of the subsequences between 0's.
I think it's easy to show that the distance between zeros is strictly
increasing,
which excludes existence of a loop,
and also implies that the asymptotic density of "0" (as well of "5"
which only occurs immediately before a "0") is zero.
The other digits seem to occur with relative densities of about
19%, 21%, 13%, 7%, 14%, 6%, 10% and 9%.
This can be computed to arbitrary precision by constructing the
limiting sequence Bob pointed out,
...,3,24,18,9,18,6,3,6,21,27,9,3,15,0.

Below are the first ~900 terms I get (structured to see the limiting
sequence ending in "0")
and some PARI code for my records.

Best,
Maximilian

(PARI)
EA(n,s=50,d=[])={for(i=1,n,print1(s",");d=concat(d,if(s,digits(s)));s=3*d[1];d=vecextract(d,"^1"));s}
digit_count(s,c=vector(57))={for(i=1,#s=Vecsmall(s),c[s[i]]++);vecextract(c,"44..")}

50,
15,0,
3,[ 15,0,]
9,[ 3,15,0,]
27,[ 9,3,15,0,]
6,21,[ 27,9,3,15,0,]
18,6,3,[ 6,21, 27,9,3,15,0,]
3,24,18,9,[ 18,6,3,6,21,27,9,3,15,0,]
9,6,12,3,24,27,[ 3,24,18,9,18,6,3,6,21,27,9,3,15,0,]

27,18,3,6,9,6,12,6,21,[
9,6,12,3,24,27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]

6,21,3,24,9,18, 27,18,3,6,18,6,3,[ 27,18,3,6,9,6,12,6,21,
9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]

18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,[
6,21,3,24,9,18,27,18,3,6,18,6,3, 27,18,3,6,9,6,12,6,21,
9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]

3,24,18,9,27,18,3,6,6,21,9,6,12,18,6,3,9,6,12,27,3,24,9,6,12,3,24,27,[
18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,
6,21,3,24,9,18,27,18,3,6,18,6,3, 27,18,3,6,9,6,12,6,21,
9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]

9,6,12,3,24,27,6,21,3,24,9,18,18,6,3,27,18,3,6,3,24,18,9,27,18,3,6,6,21,9,6,12,27,18,3,6,9,6,12,6,21,[
3,24,18,9,27,18,3,6,6,21,9,6,12,18,6,3,9,6,12,27,3,24,9,6,12,3,24,27,
18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,
6,21,3,24,9,18, 27,18,3,6,18,6,3, 27,18,3,6,9,6,12,6,21,
9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]

27,18,3,6,9,6,12,6,21,18,6,3,9,6,12,
27,3,24,3,24,18,9,6,21,3,24,9,18,9,6,12,3,24,
27,6,21,3,24,9,18,18,6,3, 27,18,3,6,6,21,3,24,9,18, 27,18,3,6,18,6,3,[
9,6,12,3,24,27,
6,21,3,24,9,18,18,6,3,27,18,3,6,3,24,18,9,27,18,3,6,6,21,9,6,12,27,18,3,6,9,6,12,6,21,
3,24,18,9,27,18,3,6,6,21,9,6,12,18,6,3,9,6,12,27,3,24,9,6,12,3,24,27,
18,6,3,9,6,12, 27,3,24,6,21,3,24,9,18,3,24,18,9,
6,21,3,24,9,18, 27,18,3,6,18,6,3,27,18,3,6,9,6,12,6,21,
9,6,12,3,24, 27,3,24,18,9,18,6,3,6,21, 27,9,3,15,0,]

6,21,3,24,9,18, 27,18,3,6,18,6,3,3,24,18,9,
27,18,3,6,6,21,9,6,12,9,6,12,3,24, 27,18,6,3,9,6,12, 27,3,24,
27,18,3,6,9,6,12,6,21,18,6,3,9,6,12,
27,3,24,3,24,18,9,6,21,3,24,9,18,18,6,3,9,6,12,
27,3,24,6,21,3,24,9,18,3,24,18,9,[
27,18,3,6,9,6,12,6,21,18,6,3,9,6,12, ..., 27,9,3,15,0,]

On Wed, Oct 1, 2014 at 9:12 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>
> Hello SeqFans,
> is T containing 10% of 0s, 10% of 1s, 10% of 2s,... 10% of 9s?
>
> T starts with 50 and is always extended with the triple of the
> leftmost digit not yet tripled in T.
>
> T = 50,15,0,3,15,0,9,3,15,0,27,9,3,15,0,6,21,27,9,3,15,0,18,6,3,6,21,27,9,3,15,0,3,24,...
> (if T enters into a loop, the question is closed -- but I can't
> find one).
>
> I would have the same question with Q: Q starts with 90 and is
> always extended with the cube of the leftmost digit not yet cubed
> in Q.
>
> Q = 90,729,0,343,8,729,0,27,64,27,512,343,8,729,0,8,343,216,64,8,343,125,1,8,27,64,27,...
>
> Best,
> É.
>
>
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/

--
Maximilian

```