In a meanwhile deleted question I had mentioned my observation, that $$H\left(2^m-1\right) = H\left(3*(2^m-1)\right) = H\left(3*(2^m-1)\ +\ 1\right)$$ 
where $H()$ denotes the Hamming weight.  
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>**Question:**  

>are there other examples of combinations of a set $\Sigma\subset\mathbb{N}$ with a function $f: \mathbb{N}\ni i\mapsto j\in\mathbb{N} $ with the following properties:  

>- the elements of $\Sigma$ can be generated from the elements of $\mathbb{N}$ via a finite sequence of arithmetic integer operations.  

>- $f$ can be evaluated with fixed, finite sequence of arithmetic integer operations  

>- $H\left(\sigma\in\Sigma\right) = H\left(f(\sigma)\right)$  

in the original observation $\Sigma:=\lbrace 2^{i+1}-1|i\in\mathbb{N}\rbrace$ and $f:=3k\ $ resp., $\ f:=3k+1$