This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384991 #14 Jun 19 2025 18:41:58 %S A384991 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 A384991 255,26,165,28,440,60,31,442,33,1386,805,154,37,105,39,1020,216,208, %U A384991 43,40,45,2860,1953,90,49,45,51,1092,120,184,55,56,150,58,6045 %N A384991 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 A384991 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 A384991 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 A384991 That is, the moved positions follow this progression: %C A384991 Position Moved Differences Between Positions %C A384991 =================== ============================= %C A384991 2 - 1 = 1 1 - 0 = 1 (= 2 - 1) %C A384991 (1) + 3 - 1 = 3 3 - 1 = 2 (= 3 - 1) %C A384991 (3) + 5 - 1 = 7 7 - 3 = 4 (= 5 - 1) %C A384991 (7) + 7 - 1 = 13 13 - 7 = 6 (= 7 - 1) %C A384991 (13) + 11 - 1 = 23 23 - 13 = 10 (= 11 - 1) %C A384991 ^^ . ^^ . %C A384991 p(k) . p(k)-1 . %C A384991 . . %C A384991 A given element can be moved multiple times before reaching its final position. %H A384991 Chuck Seggelin, <a href="/A384991/b384991.txt">Table of n, a(n) for n = 1..1000</a> %e A384991 For n=15, the rotations to construct the permutation are %e A384991 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 %e A384991 \----------------------------------------------/ 1st rotation (p=2) %e A384991 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 2 %e A384991 \----------------------------------------/ 2nd rotation (p=3) %e A384991 1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 2, 5 %e A384991 \---------------------------/ 3rd rotation (p=5) %e A384991 1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 2, 5, 10 %e A384991 \-----/ 4th rotation (p=7) %e A384991 1, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 2, 10, 5 %e A384991 The 4th rotate is an example of an element (5) which was previously rotated to the end, being rotated to the end again. %e A384991 This final permutation has order a(15) = 45 (applying it 45 times reaches the identity permutation again). %o A384991 (Python) %o A384991 from sympy.combinatorics import Permutation %o A384991 from sympy import isprime, prime %o A384991 def apply_transformation(seq): %o A384991 k = 1 %o A384991 p = prime(k) %o A384991 i = p - 1 %o A384991 while i < len(seq): %o A384991 seq.append(seq.pop(i)) %o A384991 k += 1 %o A384991 p = prime(k) %o A384991 i += (p-1) %o A384991 return seq %o A384991 def a(n): %o A384991 seq = list(range(n)) %o A384991 p = apply_transformation(seq.copy()) %o A384991 return Permutation(p).order() %Y A384991 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 A384991 nonn %O A384991 1,3 %A A384991 _Chuck Seggelin_, Jun 14 2025