Show/disprove that a system of equations expresses all the natural numbers $\forall\mathbb{N}$ I discovered the following system of equations that seem to complement each other;
I'm wondering if they are indeed equivalent to all the natural numbers $\forall\mathbb{N}$?
$$
  \begin{gather}
  D(1) = 2k + 2 \\
  D(2) = 2^2k + 1 \\
  D(3) = 2^3k + 7 \\
  D(4) = 2^4k + 3 \\
  D(5) = 2^5k + 27 \\
  D(6) = 2^6k + 11 \\
  D(7) = 2^7k + 107 \\
  D(8) = 2^8k + 43 \\
  D(9) = 2^9k + 427 \\
  \dots
  \end{gather}
$$
If we model the series as $D(n | n \in \mathbb{N}) = \{2^nk + X \mid k \ge 0\}$
The offsets $X=\{2,1,7,3,27,\dots\}$ are found from the series $2$ orders below by $4X-1$
For example $7 = (2)(4) -1$, $3 = (1)(4)-1$, $27 = (7)(4)-1$ and so on
Sketching the values seem to produce collectively all the natural numbers and without any overlaps. The diagram illustrates the first $33$ terms, showing how the series seems to complement each other

Trying to sum the series together, or by other methods, I wish to prove (or disprove) that all the numbers are in fact expressed, but not sure how to proceed. Also wondering if the system of equations are something that is recognised elsewhere.
 A: This is rather a hint than an answer...       
\begin{gather}
D(1) = 2k + 2   & \text{ all even numbers} \\
 \text{ now all odd numbers}\\
D(2) = 2^2k + 1 & \text{all odd numbers =1 modulo 4}\\
 \text{ now all odd numbers =3 modulo 4 remain and have to be listed} \\
 \text{this are that =3 modulo 8  and that =7 modulo 8 } \\
 \text{ first that of =7 modulo 8} \\
D(3) = 2^3k + 7\\
 \text{ now remaining =3 and = 11 modulo 16} \\
D(4) = 2^4k + 3 & \text{ and so on} \\
\dots
\end{gather}                               
I think you can decode this/formalize this now yourself... What you need is a formalization for the residues which satisfy that in each step 2 exclusive classes of numbers remain. I've seen this in the context of the Collatz-problem 2.3,pg 6-7 where there are two sequences of residues $1,5,21,85,...$ and $3,13,53,213,...$ which provide a similar classification of the natural numbers.
(For instance I've seen, that your residues follow two interwoven sequences with the rule $a_{k+1}=4a_k-1$)
