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A339491 Lexicographically earliest longest simple path in the divisor graph of {1,...,n}. Irregular triangle read by rows. 6
1, 1, 2, 2, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 3, 6, 2, 4, 1, 5, 3, 6, 2, 4, 1, 5, 3, 6, 2, 4, 8, 1, 5, 4, 8, 2, 6, 3, 9, 1, 5, 4, 8, 1, 5, 10, 2, 6, 3, 9, 4, 8, 1, 5, 10, 2, 6, 3, 9, 5, 10, 2, 8, 4, 12, 6, 3, 9, 1, 7, 5, 10, 2, 8, 4, 12, 6, 3, 9, 1, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A simple path in the divisor graph of {1,...,n} is a sequence of distinct numbers between 1 and n such that if k immediately follows m, then either k divides m or m divides k. For more information, references and links see A337125.

LINKS

Table of n, a(n) for n=1..81.

EXAMPLE

1:                     [1],

2:                    [1, 2],

3:                  [2, 1, 3],

4:                 [2, 4, 1, 3],

5:                 [2, 4, 1, 3],

6:              [3, 6, 2, 4, 1, 5],

7:              [3, 6, 2, 4, 1, 5],

8:             [3, 6, 2, 4, 8, 1, 5],

9:            [4, 8, 2, 6, 3, 9, 1, 5],

10:         [4, 8, 1, 5, 10, 2, 6, 3, 9],

11:         [4, 8, 1, 5, 10, 2, 6, 3, 9],

12:      [5, 10, 2, 8, 4, 12, 6, 3, 9, 1, 7],

13:      [5, 10, 2, 8, 4, 12, 6, 3, 9, 1, 7],

14:     [5, 10, 1, 7, 14, 2, 8, 4, 12, 6, 3, 9],

15:   [6, 12, 4, 8, 1, 7, 14, 2, 10, 5, 15, 3, 9],

16:  [6, 12, 4, 8, 16, 1, 7, 14, 2, 10, 5, 15, 3, 9].

MAPLE

with(Iterator):

DivisorPath := proc(n, k) local c, p, w, isok;

    isok := proc(A) local e, i, di; e := nops(A) - 1;

       di := (n, k) -> evalb(irem(n, k) = 0 or irem(k, n) = 0):

       for i from 1 to e while di(A[i], A[i+1]) do od;

       return evalb(i = e + 1) end:

    for c in Combination(n, k) do

       for p in Permute([seq(j + 1, j in c)], k) do

           w := convert(p, list);

           if isok(w) then return w fi:

od  od  end:

A337125 := [1, 2, 3, 4, 4, 6, 6, 7, 8, 9, 9]:

for n from 1 to 9 do DivisorPath(n, A337125[n]) od;

CROSSREFS

Cf. A337125 (row length), A339490.

Cf. A339492, A339489, A339496.

Cf. A340114 (a variant problem).

Sequence in context: A159803 A308934 A058741 * A291712 A074945 A323479

Adjacent sequences:  A339488 A339489 A339490 * A339492 A339493 A339494

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Dec 29 2020

EXTENSIONS

Signposting added to first comment by Peter Munn, Mar 12 2021

STATUS

approved

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Last modified May 9 13:42 EDT 2021. Contains 343742 sequences. (Running on oeis4.)