[seqfan] A sequence and a few questions

[Ali Sada]

Hi everyone,

Start with a list of positive integers 1,2,3,4… Take each number in the list (m) in turn and repeat it m times. Place the repeated numbers in non-consecutive locations to the right of m and amend the list accordingly.
We start with 
1,2,3,4,5,6,7,8,9,10…
We repeat the 1 one time and put it after 2.
1,2,1,3,4,5,6,7,8…
We repeat the 2 twice and place the new “copies” after the second 1 and the first 3
1,2,1,2,3,2,4,5,6,7,8….
We repeat the 3 three times and place the new repeats after the first 3 such that they are not in consecutive positions
1,2,1,2,3,2,3,4,3,5,3,6,7,8…
And so on. We get: 

 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 3, 5, 4, 5, 6, 5, 6, 4, 6, 5, 6, 7, 6, 7, 8, 7, 8, 6, 8, 7, 8… 

1.    Is this sequence suitable for the OEIS? 
2.    Is the definition above accurate? 
3.    Can we find a general formula?
4.    If we tweak the definition a little bit: “Start with a list of positive integers 1,2,3,4… Take each number in the list (m) in turn and repeat it m times. Place the repeated numbers in non-consecutive locations such that the first appearance of m occurs at n = m.
1,2,3,4,5,6,7,8…
1,2,1,3,4,5,6,7…
1,2,1,3,2,4,5,6,7…
1,2,3,1,3,2,3,4,3,5,6,7…
1,2,3,4,1,4,3,4,2,4,3,4,5…

Gradually we will have 1,2,3,4,5,6,7…
Is this sequence equal to A000027 (regardless of the fact that there are countless numbers that are supposed to appear “later”)? 

Best,

Ali