A067957 Number of divisor chains of length n: permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

1, 1, 1, 2, 2, 4, 5, 7, 7, 24, 22, 29, 39, 67, 55, 386, 235, 312, 347, 451, 1319, 5320, 3220, 4489, 20237, 36580, 52875, 197103, 216562, 289478, 567396, 659647, 1111153, 3131774, 2200426, 29523302, 34214028, 48161995, 32616148, 242860900, 293579041, 363415618

Offset: 0

Author

Floor van Lamoen, Mar 06 2002

Keywords

• nonn

Comments

Apparently this sequence originated in a problem composed by Matthijs Coster in 2002.

Let M = floor(n/2), then the following permutations always work: for n even: M+1, 1, M+2, 2, ..., n-1, M-1, n, M; for n odd: M+1, 1, M+2, 2, ..., M-1, n-1, M, n. - Daniel Asimov, May 04 2004

Examples

			Examples of divisor chains of lengths 1 through 9:
  1
  2 1
  3 1 2
  4 2 3 1
  5 1 2 4 3
  6 2 4 3 5 1
  7 1 2 5 3 6 4
  8 2 5 3 6 4 7 1
  8 4 3 5 1 7 2 6 9
The five divisor chains of length 6 are:
  4 1 5 2 6 3
  4 2 6 3 5 1
  5 1 2 4 6 3
  5 1 6 4 2 3
  6 2 4 3 5 1. - Eugene McDonnell, May 21 2004
		

Links

• Matthijs Coster, Sequences
• Matthijs Coster, Problem 2001/3-A of the Universitaire Wiskunde Competitie, Nieuw Arch. Wisk. 5/3 (2002), 92-94.

Crossrefs

Cf. A093313, A093314, A093315, A094097, A094098, A094099.

Extensions

a(31)-a(35) from Jud McCranie, May 06 2004

a(0)=1 prepended by Alois P. Heinz, Aug 26 2017

a(36)-a(41) from Zhao Hui Du, May 12 2024