A360268 A version of the Josephus problem: a(n) is the surviving integer under the following elimination process. Arrange 1,2,3,...,n in a circle, increasing clockwise. Starting with i=1, delete the integer 5 places clockwise from i. Repeat, counting 5 places from the next undeleted integer, until only one integer remains.

1, 1, 1, 3, 4, 4, 3, 1, 7, 3, 9, 3, 9, 1, 7, 13, 2, 8, 14, 20, 5, 11, 17, 23, 4, 10, 16, 22, 28, 4, 10, 16, 22, 28, 34, 4, 10, 16, 22, 28, 34, 40, 3, 9, 15, 21, 27, 33, 39, 45, 51, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 1, 7, 13, 19, 25

Offset

1,4

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Eric Huang, Tanya Khovanova, Timur Kilybayev, Ryan Li, Brandon Ni, Leone Seidel, Samarth Sharma, Nathan Sheffield, Vivek Varanasi, Alice Yin, Boya Yun, and William Zelevinsky, Card Dealing Math, arXiv:2509.11395 [math.NT], 2025. See p. 21.

Wikipedia, Josephus problem.

Index entries for sequences related to the Josephus Problem.

Formula

a(n) = (a(n-1) + 5) mod n + 1 if n > 1, a(1) = 1.

Example

a(7) = 3 because the elimination process gives (^1,2,3,4,5,6,7) -> (1,2,3,4,5,^7) -> (1,2,3,4,^7) -> (^1,2,3,4) -> (1,^3,4) -> (^3,4) -> (3), where ^ denotes the counting reference position.
a(13) = 9 => a(14) = (a(13) + 5) mod 14 + 1 = 1.

Mathematica

a[1] = 1; a[n_] := a[n] = Mod[a[n - 1] + 5, n] + 1; Array[a, 100] (* Amiram Eldar, Feb 03 2023 *)

Prog

(Python)
def A360268_up_to_n(n):
  val = 1
  return [val := (val + 5) % i + 1 for i in range(1, n+1)]

Crossrefs

6th column of A198789.

Cf. A006257, A054995, A088333, A181281.

Sequence in context: A073498 A105736 A248158 * A090283 A318705 A334492

Adjacent sequences: A360265 A360266 A360267 * A360269 A360270 A360271

Keyword

nonn,changed

Author

Benjamin Lilley, Jan 31 2023

Status

approved