Let $a=a_1, a_2, a_3 \ldots$ with $a_1<a_2<a_3\ldots$ such that $$\sum_{n=1}^\infty3^n2^{a_n}\equiv1 \pmod{2^\infty}$$ Let $b$ be the differences between subsequent terms of the series, such that $b_n=a_{n+1}-a_n$.
I.e., the first five values of $a$ are $0,1,2,7,8$ and the corresponding values of $b$ are $1,1,5,1$. The sum of the first five values of $3^n2^{a_n}$ are $3*1+9*2+27*4+81*128+243*256=3073\equiv1 \pmod{2^{10}}$.
My question is, is the sequence $b$ eventually periodic? And whether $b$ is periodic or not, is there a closed form for the sum $s=\sum_{n=1}^\infty2^{a_n}$?
This is equivalent to the following:
starting with $x_0 = 1$, we define $y_n = x_n - 3^n$, $b_n = v_2(y_n)$ and $x_{n + 1} = y_n/2^{b_n}$.
As far as I can tell, this is more or less just a "random" sequence. There is no reason that the sequence $b_n$ is eventually cyclic. Neither is there reason that the sum $\sum_{n = 1}^\infty 2^{a_n}$ has a closed form.
The first $400$ terms of $b_n$ are:
1, 1, 5, 1, 2, 5, 2, 1, 3, 1, 2, 4, 1, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 7, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 5, 3, 2, 1, 1, 1, 3, 1, 3, 2, 1, 1, 2, 1, 3, 1, 3, 2, 3, 2, 6, 1, 4, 4, 1, 2, 3, 1, 1, 1, 4, 1, 3, 2, 2, 1, 5, 2, 2, 1, 1, 1, 2, 1, 2, 5, 3, 1, 1, 3, 3, 1, 2, 1, 1, 1, 3, 3, 4, 1, 1, 2, 4, 2, 1, 3, 1, 3, 1, 1, 1, 1, 1, 4, 3, 1, 5, 3, 1, 4, 1, 1, 2, 1, 3, 1, 2, 3, 4, 1, 3, 3, 2, 1, 2, 3, 6, 1, 1, 2, 2, 4, 1, 2, 4, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 3, 1, 2, 3, 1, 2, 1, 7, 2, 1, 5, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 3, 2, 3, 2, 3, 2, 2, 4, 1, 1, 1, 2, 1, 4, 1, 4, 2, 2, 1, 3, 1, 3, 1, 1, 1, 3, 2, 2, 2, 1, 1, 2, 3, 3, 2, 1, 3, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 2, 1, 1, 1, 1, 3, 4, 2, 1, 1, 5, 1, 1, 2, 2, 3, 2, 1, 1, 1, 3, 2, 1, 3, 3, 4, 3, 1, 2, 5, 2, 1, 2, 2, 1, 1, 3, 4, 1, 2, 2, 3, 3, 1, 1, 3, 5, 3, 2, 2, 3, 1, 3, 3, 1, 3, 3, 1, 1, 1, 1, 2, 7, 1, 1, 6, 7, 3, 2, 6, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 4, 3, 4, 4, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 3, 5, 1, 2, 2, 8, 3, 1, 2, 2, 1, 1, 3, 2, 6, 3, 1, 1, 2, 1, 1, 1, 1, 2, 5, 2, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 4, 1, 5, 3, 3, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 2, 1, 6, 3