# [seqfan] Re: A problem for digit sum of n in base 3

Tue Apr 3 18:43:08 CEST 2012

```Thanks Jens, my full handy calculations were very restricted (up to n=9) and after that I calculated the iterations only in cases when  n-1 is Fibonacci.  I was so surprised that for such values of n we always have the stabilization that  thought  that the problem would be interesting for wide circle of the readers.  Peter Moses sent me more numbers answering to the posed problem in the form: the  list of those n for which we have NOT a stabilization:

{5,7,8,17,21,23,26,31,39,40,41,45,49,51,52,53,58,62,64,67,69,78,81,82,84,87,91,93,108,113,115,
116,119,121,122,128,131,135,136,139,142,151,152,..}
and we are going to submit this sequence to OEIS.

Best regards,

----- Original Message -----
From: jens at voss-ahrensburg.de
Date: Tuesday, April 3, 2012 15:22
Subject: [seqfan] Re: A problem for digit sum of n in base 3
To: seqfan at list.seqfan.eu

>
> Sorry, Vladimir, but this seems to be a case of the
> http://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers:
>
> For n=10, s(1024) = 8, s(1032) = 6, s(1030) = 6, ... (stabilization),
> but n-1 = 9 is not a Fibonacci number.
>
> Regards,
> Jens
>
> > Dear SeqFans,
> >
> > Let s(n)=s_3(n) be digit sum of n in base 3. Consider
> iterations: a_1(n)
> =s
> > (2^n), a_2(n)=s(2^n+a_1(n)),
> > a_3(n)=s(2^n+a_2(n)),...
> > Question. For which n there exists N=N(n) such that, for k>N, a_k(n)
> =consta
> > nt(k)?
> > It is interesting that for a few small n such a stabilization
> arises
> only w
> > hen n-1 is a FIBONACCI number.
> > I am not sure that it is kept for larger n. If anyone can
> verify that?
> > Examples. For n=5, s(32)=4, s(36)=2, s(34)=4, s(36)=2,...
> (without
> stabiliz
> > ation);
> >                 For n=6, s(64)=4, s(68)=6, s(70)=6,
> s(70)=6,... (st
> > abilization)
> >
> > Regards,
> >
> >
> >
> > _______________________________________________
> >
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