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.

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 and doesn't amount to a mere shift operation, i.e. $f(n)\ne 2^kn$

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

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

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.

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 and doesn't amount to a mere shift operation, i.e. $f(n)\ne 2^kn$

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

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

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.

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., $\ f:=3k+1$

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.

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 and doesn't amount to a mere shift operation, i.e. $f(n)\ne 2^kn$

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

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