A093314 -id:A093314 - cp's OEIS frontend

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

Floor van Lamoen, Mar 06 2002

nonn

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 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

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

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

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

A093313 Number of permutations s_1,s_2,...,s_n of 1,2,...,n with s_1 = 2 and such that for all j=1,2,...,n, s_j divides Sum_{i=1..j} s_i.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 1, 11, 9, 15, 14, 14, 23

Offset: 1

Matthijs Coster, Apr 26 2004; revised Aug 05 2005

An easy calculation turns out that the beginning elements are always: 2,1,3,6,(then either 4 or 12),...

The total number of permutations with this property is given in A067957.

			There is a unique permutation of the numbers 1..38, starting with 2, namely:
2 1 3 6 12 24 8 28 21 35 14 22 4 20 25 5 23 11 33 27 9 37 10 19 7 29 15 30 16 31 17 32 36 34 38 18 26 13
with corresponding sums
2 3 6 12 24 48 56 84 105 140 154 176 180 200 225 230 253 264 297 324 333 370 380 399 406 435 450 480 496 527 544 576 612 646 684 702 728 741.

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

Cf. A067957, A093314, A093315.

cp's OEIS Frontend

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.

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