1,3

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.

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.

That is, the moved positions follow this progression:

Position Moved Differences Between Positions

=================== =============================

2 - 1 = 1 1 - 0 = 1 (= 2 - 1)

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

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

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

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

^^ . ^^ .

p(k) . p(k)-1 .

. .

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

Chuck Seggelin, Table of n, a(n) for n = 1..1000

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

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

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

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

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

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

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

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

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

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

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

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

(Python)

from sympy.combinatorics import Permutation

from sympy import isprime, prime

def apply_transformation(seq):

k = 1

p = prime(k)

i = p - 1

while i < len(seq):

seq.append(seq.pop(i))

k += 1

p = prime(k)

i += (p-1)

return seq

def a(n):

seq = list(range(n))

p = apply_transformation(seq.copy())

return Permutation(p).order()

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).

Sequence in context: A029579 A106647 A130157 * A265018 A238792 A158745

Adjacent sequences: A384988 A384989 A384990 * A384992 A384993 A384994

Chuck Seggelin, Jun 14 2025

approved