OFFSET
1,6
COMMENTS
LINKS
Peter Kagey, Table of n, a(n) for n = 1..10000
Peter Kagey, Math.StackExchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?
Wikipedia, Sperner property of a partially ordered set
EXAMPLE
For n=15, the paths are shown vertically at left, and the graph obtained appears at right:
15 15 15 15 15 => 15
| | | | | _/ \_
| | | | | / \
10 10 12 12 12 => 10 12
| | | | | | \_ _/ |
| | | | | | \ / |
5 8 6 6 8 => 5 8 6
| | | | | \_ | _/|
| | | | | \_|_/ |
4 4 3 4 4 => 4 3
| | | | | | _/
| | | | | |_/
2 2 2 2 2 => 2
| | | | | |
| | | | | |
1 1 1 1 1 => 1
Because the maximum number of distinct terms in any row is 3, a(15) = 3.
MATHEMATICA
Max[Length@ Union@ # & /@ Transpose@ #] & /@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 105]
(* Second program: *)
g[n_] := Block[{lst = {{n}}}, While[lst[[-1]] != {1}, lst = Join[lst, {Union@ Flatten[# - #/(First@ # & /@ FactorInteger@ #) & /@ lst[[-1]] ]}]]; Max[Length /@ lst]]; Array[g, 105] (* Robert G. Wilson v, May 08 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved