%I #14 Jun 19 2025 18:41:58

%S 1,1,2,3,3,5,4,7,8,15,10,11,12,13,45,15,105,17,77,19,24,21,117,23,504,

%T 255,26,165,28,440,60,31,442,33,1386,805,154,37,105,39,1020,216,208,

%U 43,40,45,2860,1953,90,49,45,51,1092,120,184,55,56,150,58,6045

%N Order of the permutation of [n] formed by a Josephus elimination variation: take p, skip 1, with p starting at 2 and advancing to the next prime after each skip.

%C The Josephus elimination begins with a circular list [n] from which successively take p elements and skip 1 where p begins at 2 and increases to the next prime (3,5,7,11,13,...) after each skip, and the permutation is the elements taken in the order they're taken.

%C Let p(k) be the k-th prime number, and let k increment with each move. In this variation, "take p(k), skip 1" means: move the p(k)-th element to the end of the list. After each move, counting begins from the element that replaced the moved one, and the next move targets the subsequent p(k+1)-th element. Thus, the positions of the elements being moved are p(k+1)-1 apart.

%C That is, the moved positions follow this progression:

%C Position Moved Differences Between Positions

%C =================== =============================

%C 2 - 1 = 1 1 - 0 = 1 (= 2 - 1)

%C (1) + 3 - 1 = 3 3 - 1 = 2 (= 3 - 1)

%C (3) + 5 - 1 = 7 7 - 3 = 4 (= 5 - 1)

%C (7) + 7 - 1 = 13 13 - 7 = 6 (= 7 - 1)

%C (13) + 11 - 1 = 23 23 - 13 = 10 (= 11 - 1)

%C ^^ . ^^ .

%C p(k) . p(k)-1 .

%C . .

%C A given element can be moved multiple times before reaching its final position.

%H Chuck Seggelin, <a href="/A384991/b384991.txt">Table of n, a(n) for n = 1..1000</a>

%e For n=15, the rotations to construct the permutation are

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

%e \----------------------------------------------/ 1st rotation (p=2)

%e 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 2

%e \----------------------------------------/ 2nd rotation (p=3)

%e 1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 2, 5

%e \---------------------------/ 3rd rotation (p=5)

%e 1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 2, 5, 10

%e \-----/ 4th rotation (p=7)

%e 1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 2, 10, 5

%e The 4th rotate is an example of an element (5) which was previously rotated to the end, being rotated to the end again.

%e This final permutation has order a(15) = 45 (applying it 45 times reaches the identity permutation again).

%o (Python)

%o from sympy.combinatorics import Permutation

%o from sympy import isprime, prime

%o def apply_transformation(seq):

%o k = 1

%o p = prime(k)

%o i = p - 1

%o while i < len(seq):

%o seq.append(seq.pop(i))

%o k += 1

%o p = prime(k)

%o i += (p-1)

%o return seq

%o def a(n):

%o seq = list(range(n))

%o p = apply_transformation(seq.copy())

%o return Permutation(p).order()

%Y Cf. A000040, A051732 (Josephus elimination permutation order), A384753 (take 2 skip 1 Josephus variation), A384989 (take 3 skip 1 Josephus variation), A384990 (take k skip 1 Josephus variation).

%K nonn

%O 1,3

%A _Chuck Seggelin_, Jun 14 2025