[seqfan] The Right Arm Reach Sequence

[Ali Sada]

Hi everyone,

Each positive integer k has k “right arm reach”. And numbers between k+1 and 2k cannot be within k's right arm reach. 
We start with a(1) = 1. 
Now, 2 cannot follow directly because the second position is within the right arm reach of 1. So, a(2) = 3.
This step will send 4,5, and 6 to the “bullpen” waiting for their turns, while 2 can go back to the field since it’s out of 1’s right arm reach now. a(3) = 2. 
After that, the least positive integer that’s not in the bullpen is 7, so, a(4) = 7.  This step will send all the numbers between 8 and 14 to the bullpen. Since we are now in the 3’s and 7’s right arm reaches, the least available positive integer is 15. So, a(5) = 15. And now we are outside of the 3’s right arm reach, so, 4 comes out of the bullpen and a(6) = 4. And so on. 
Is the definition below accurate? And can you please help find the terms of this sequence? 
“Starting at a(1) = 1, a(n) is the least positive integer, not already in the sequence, that satisfies the following condition for all existed terms: numbers between a(m)+1 and 2a(m) cannot be positioned at any location between m+1 and m + a(m).”

Best,

Ali