cp's OEIS Frontend

A384990 Order of the permutation of [n] formed by a Josephus elimination variation: take k, skip 1, with k starting at 1 and increasing by 1 after each skip.

%I A384990 #9 Jun 19 2025 18:37:40
%S A384990 1,1,2,2,4,5,6,7,15,9,12,11,12,13,14,60,16,70,24,88,20,60,22,23,24,25,
%T A384990 26,27,420,29,221,31,3465,33,285,35,840,37,38,1040,40,41,2618,43,44,
%U A384990 2520,46,546,48,594,840,644,52,696,54,2520,56,57,58,59,60,61,62
%N A384990 Order of the permutation of [n] formed by a Josephus elimination variation: take k, skip 1, with k starting at 1 and increasing by 1 after each skip.
%C A384990 The Josephus elimination begins with a circular list [n] from which successively take k elements and skip 1 where k begins at 1 and monotonically increases after each skip, and the permutation is the elements taken in the order they're taken.
%C A384990 Take k and move 1 is a move every k-th element, but with the next k+1 elements reckoned inclusive of the element which replaced the moved 1, and hence positions k apart.
%C A384990 A given element can be skipped or moved multiple times before reaching its final position.
%C A384990 This sequence enters relatively lengthy stretches of linearity where a(n) = n-1 before entering stretches where it oscillates between n-1 and much larger values.  This behavior is observed multiple times between a(1) and a(1000).  It is unknown if this behavior continues to happen further into the sequence.  For example: a(n)=n-1 for n=905 to 946, and the terms that follow are 9419588158802421600, 947, 224555, 949, 1582305192, 951, 226455, 953, etc.
%e A384990 For n=10, the rotations to construct the permutation are
%e A384990     1, 2, 3, 4, 5, 6, 7, 8, 9, 10
%e A384990        \--------------------------/  1st rotation (k=1)
%e A384990     1, 3, 4, 5, 6, 7, 8, 9, 10, 2
%e A384990           \-----------------------/  2nd rotation (k=2)
%e A384990     1, 3, 5, 6, 7, 8, 9, 10, 2, 4
%e A384990                 \-----------------/  3rd rotation (k=3)
%e A384990     1, 3, 5, 6, 8, 9, 10, 2, 4, 7
%e A384990                           \-------/  4th rotation (k=4)
%e A384990     1, 3, 5, 6, 8, 9, 10, 4, 7, 2
%e A384990 The 4th rotate is an example of an element (2) which was previously rotated to the end, being rotated to the end again.
%e A384990 This final permutation has order a(10) = 9 (applying it 9 times reaches the identity permutation again).
%o A384990 (Python)
%o A384990 from sympy.combinatorics import Permutation
%o A384990 def apply_transformation(seq):
%o A384990     step = 1
%o A384990     i = step
%o A384990     while i < len(seq):
%o A384990         seq.append(seq.pop(i))
%o A384990         step += 1
%o A384990         i += step - 1
%o A384990     return seq
%o A384990 def a(n):
%o A384990     seq = list(range(n))
%o A384990     p = apply_transformation(seq.copy())
%o A384990     return Permutation(p).order()
%Y A384990 Cf. A051732 (Josephus elimination permutation order), A384753 (take 2 skip 1 Josephus variation), A384989 (take 3 skip 1 Josephus variation).
%K A384990 nonn
%O A384990 1,3
%A A384990 _Chuck Seggelin_, Jun 14 2025