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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
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.
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. A340114 (a variant problem).
Sequence in context: A159803 A308934 A058741 * A346871 A375300 A291712
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Dec 29 2020
EXTENSIONS
Signposting added to first comment by Peter Munn, Mar 12 2021
STATUS
approved