A198790 Irregular table T(n,k) read by rows: Last survivor positions in Josephus problem for n numbers and a count of k, n >= 1, lcm(1, 2, 3, ..., n) >= k >= 1.

1, 2, 1, 3, 3, 2, 2, 1, 1, 4, 1, 1, 2, 2, 3, 2, 3, 3, 4, 4, 1, 5, 3, 4, 1, 2, 4, 4, 1, 2, 4, 5, 3, 2, 5, 1, 3, 4, 1, 1, 3, 4, 1, 2, 5, 4, 2, 3, 5, 1, 3, 3, 5, 1, 3, 4, 2, 1, 4, 5, 2, 3, 5, 5, 2, 3, 5, 1, 4, 3, 1, 2, 4, 5, 2, 2, 4, 5, 2, 3, 1, 6, 5, 1, 5, 1, 4

Offset

1,2

Comments

Arrange 1, 2, 3, ... n clockwise in a circle. Starting the count at 1, delete every k-th integer clockwise until only one remains, which is T(n,k).

In the full table in A198789, row n repeats with a periodicity of lcm(1, 2, 3, ..., n) = A003418(n). This sequence is a scan of each row in A198789 for exactly one period length.

Links

William Rex Marshall, Rows n = 1..10, flattened

Index entries for sequences related to the Josephus Problem

Formula

T(1,1) = 1;

for n >= 2, lcm(1, 2, ... n) >= k >=1: T(n,k) = ((T(n-1,((k-1) mod lcm(1, 2, ... n-1)) + 1) + k - 1) mod n) + 1.

Example

n\k  1  2  3  4  5  6  7  8  9 10 11 12
---------------------------------------
1 |  1
2 |  2  1
3 |  3  3  2  2  1  1
4 |  4  1  1  2  2  3  2  3  3  4  4  1

Crossrefs

Cf. A003418, A007495, A032434, A198788, A198789.

Sequence in context: A241534 A337137 A035050 * A306995 A212907 A308584

Adjacent sequences: A198787 A198788 A198789 * A198791 A198792 A198793

Keyword

nonn,easy,tabf

Author

William Rex Marshall, Nov 21 2011

Status

approved