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

[Vladimir Shevelev]

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)=constant(k)?
It is interesting that for a few small n such a stabilization arises only when 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 stabilization);
                For n=6, s(64)=4, s(68)=6, s(70)=6, s(70)=6,... (stabilization)

Regards,
Vladimir


 Shevelev Vladimir‎