A093314 -id:A093314

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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.

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

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

LINKS

Table of n, a(n) for n=0..41.

Matthijs Coster, Sequences

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

EXAMPLE

Examples of divisor chains of lengths 1 through 9:

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Floor van Lamoen, Mar 06 2002

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

STATUS

approved

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,36

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..44.

Matthijs Coster, Sequences

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

EXAMPLE

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.

CROSSREFS

Cf. A067957, A093314, A093315.

KEYWORD

nonn,more

AUTHOR

Matthijs Coster, Apr 26 2004; revised Aug 05 2005

STATUS

approved

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