A363584 - OEIS

%I #23 Aug 24 2023 10:26:36

%S 1,3,2,7,15,4,31,63,127,255,5,11,511,1023,2047,4095,6,8191,16383,

%T 32767,23,65535,131071,8,17,262143,524287,1048575,2097151,4194303,

%U 8388607,16777215,9,33554431,67108863,134217727,268435455,536870911,47,1073741823,2147483647

%N Lexicographically least sequence of distinct positive integers, subject to the constraint that if 1 <= k <= a(m), then either a(m+k) < a(m) or a(m+k) > 2*a(m).

%C This sequence was called the Right Arm Reach sequence in a Seqfan thread.

%C Every positive integer appears, and the index at which k appears is at most k*(k+1)/2. - _Pontus von Brömssen_, Aug 22 2023

%e 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.

%o (Python)

%o from itertools import islice

%o def A363584_generator():

%o     alist = []

%o     while True:

%o         x = 1

%o         while True:

%o             for i,a in enumerate(alist):

%o                 if a <= x <= 2*a and i <= a-1:

%o                     x = 2*a+1

%o                     break

%o                 if x == a:

%o                     x += 1

%o                     break

%o             else: break

%o         yield x

%o         alist.insert(0,x)

%o def A363584_list(nterms):

%o     return list(islice(A363584_generator(),nterms)) # _Pontus von Brömssen_, Aug 19 2023

%K nonn

%O 1,2

%A _Ali Sada_, Aug 17 2023

%E More terms from, and name clarified by _Pontus von Brömssen_, Aug 22 2023