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%I #23 May 08 2020 23:21:01
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C Maximum term in row n of A334184.
%H Peter Kagey, <a href="/A334144/b334144.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Kagey, Math.StackExchange, <a href="https://math.stackexchange.com/questions/3632156/does-a-graded-poset-on-mathbbn-0-generated-from-subtracting-factors-defi">Does a graded poset on the positive integers generated from subtracting factors define a lattice?</a>
%H Richard Stanley, <a href="https://ocw.mit.edu/courses/mathematics/18-318-topics-in-algebraic-combinatorics-spring-2006/lecture-notes/sperner.pdf">MIT OpenCourseWare, Lecture Notes in Algebraic Combinatorics: The Sperner Property</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sperner_property_of_a_partially_ordered_set">Sperner property of a partially ordered set</a>
%e For n=15, the paths are shown vertically at left, and the graph obtained appears at right:
%e 15 15 15 15 15 => 15
%e | | | | | _/ \_
%e | | | | | / \
%e 10 10 12 12 12 => 10 12
%e | | | | | | \_ _/ |
%e | | | | | | \ / |
%e 5 8 6 6 8 => 5 8 6
%e | | | | | \_ | _/|
%e | | | | | \_|_/ |
%e 4 4 3 4 4 => 4 3
%e | | | | | | _/
%e | | | | | |_/
%e 2 2 2 2 2 => 2
%e | | | | | |
%e | | | | | |
%e 1 1 1 1 1 => 1
%e Because the maximum number of distinct terms in any row is 3, a(15) = 3.
%t 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]
%t (* Second program: *)
%t 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 *)
%Y Cf. A064097, A332992, A332999, A333123, A334111, A334184.
%Y Cf. also A096825.
%K nonn
%O 1,6
%A _Michael De Vlieger_, _Peter Kagey_, _Antti Karttunen_, Apr 15 2020
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