Can someone tell me why this happens?

I observed this awkward repetition while making a strategy for the game "odd-eve". Can someone tell me why this happens?

Let ${ a }_{ 1 }$ be the sum of digits of $x$ in base 10.

Let ${ a }_{ n}$ be the sum of digits of ${ a }_{ n-1}$ in base 10, for integers $n>1$.

Let $g\left( x \right) =\lim _{ n\rightarrow \infty }{ { a }_{ n } }$

I observed that:-

$g\left( 2 \right) =2$ $g\left( 4 \right) =4$ $g\left( 8 \right) =8$ $g\left( 16\right) =7$ $g\left( 32 \right) =5$ $g\left( 64 \right) =1$

then

$g\left( 128 \right) =2$ $g\left( 256\right) =4$ $g\left( 512 \right) =8$ $g\left( 1024 \right) =7$ $g\left( 2048 \right) =5$ $g\left( 4096 \right) =1$

and this pattern is repeating again and again.

$g\left( 8192 \right) =2$ $g\left( 16384 \right) =4$

and so on.........

So, The pattern for powers of 2 is $2,4,8,7,5,1,2,4,8,7,5,1$

and

The pattern for powers of 3 is $3,9,9,9,9,9,9,9,9,9,9,9$

The pattern for powers of 4 is $4,7,1,4,7,1,4,7,1,4,7,1$

The pattern for powers of 5 is $5,7,8,4,2,1,5,7,8,4,2,..$

Why does it work?

Does it work for powers of all integers?

Does it work in all bases?