A334144 Consider the mapping k -> (k - (k/p)), where prime p | k. a(n) = maximum distinct terms at any position j among the various paths to 1.

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 1, 4, 2, 4, 3, 3, 3, 3, 2, 2, 4, 4, 3, 4, 3, 3, 2, 4, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 1, 5, 5, 5, 2, 5, 5, 5, 3, 3, 3, 4, 3, 6, 4, 4, 2, 3, 2, 2, 4, 3, 4, 4, 3, 3, 5, 5, 3, 5, 3, 5, 2, 2, 4, 6, 3, 3, 3, 3, 3, 6, 3

Offset

1,6

Comments

Let i = A064097(n) be the common path length and let 1 <= j <= i. Given a path P, we find for any j relatively few distinct values. Regarding a common path length i, see A333123 comment 2, and proof at A064097.

Maximum term in row n of A334184.

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?

Richard Stanley, MIT OpenCourseWare, Lecture Notes in Algebraic Combinatorics: The Sperner Property

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

Cf. A064097, A332992, A332999, A333123, A334111, A334184.

Cf. also A096825.

Sequence in context: A023568 A081753 A332999 * A232551 A367006 A261129

Adjacent sequences: A334141 A334142 A334143 * A334145 A334146 A334147

Keyword

nonn

Author

Michael De Vlieger, Peter Kagey, Antti Karttunen, Apr 15 2020

Status

approved