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A334184 Irregular table read by rows: T(n,k) gives the number of values that can be reached after exactly k iterations of maps of the form (n - n/p) where p is a prime divisor of n. 0 <= k < A073933(n).

%I A334184 #36 Apr 28 2020 05:58:12
%S A334184 1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,2,1,
%T A334184 1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,2,2,1,1,1,2,2,2,1,1,1,2,3,2,1,1,1,1,1,
%U A334184 1,1,1,1,1,1,1,1,1,2,2,2,1,1,1,1,2,2,2
%N A334184 Irregular table read by rows: T(n,k) gives the number of values that can be reached after exactly k iterations of maps of the form (n - n/p) where p is a prime divisor of n. 0 <= k < A073933(n).
%C A334184 Row lengths are given by A073933(n). Row sums are given by A332809(n). The maximum value in each row is given by A334144(n).
%C A334184 The n-th row consists of all 1's if and only if n is a power of two (A000079) or a Fermat prime (A019434).
%C A334184 Conjecture: rows are unimodal (increasing and then decreasing).
%C A334184 Not all rows are unimodal. Indices of rows that have terms that increase and decrease more than once are A334238. - _Michael De Vlieger_, Apr 18 2020
%H A334184 Michael De Vlieger, <a href="/A334184/b334184.txt">Table of n, a(n) for n = 1..12386</a> (rows 1 <= n <= 1000, flattened)
%H A334184 Michael De Vlieger, <a href="/A334184/a334184.png">Hasse diagrams</a> showing rows n = {55, 63, 171, ...} that increase and decrease more than once.
%H A334184 Michael De Vlieger, <a href="/A334184/a334184.txt">Table of n, b(n)</a> for n = 1..10000, encoding the running total of row n of this sequence as a binary number expressed decimally.
%F A334184 T(n,0) = T(n, A073933(n) - 2) = T(n, A073933(n) - 1) = 1.
%F A334184 T(n,1) = A001221(n) for n > 1.
%e A334184 For n = 15, the fifteenth row of this table is [1,2,3,2,1,1] because there is one value (15 itself) that can be reached with zero iterations of (n - n/p) maps, two values (10 and 12) that can be reached after one iteration, three values (5, 8, and 6) that can be reached after two iterations, and so on.
%e A334184       15
%e A334184      _/ \_
%e A334184     /     \
%e A334184   10       12
%e A334184   | \_   _/ |
%e A334184   |   \ /   |
%e A334184   5    8    6
%e A334184    \_  |  _/|
%e A334184      \_|_/  |
%e A334184        4    3
%e A334184        |  _/
%e A334184        |_/
%e A334184        2
%e A334184        |
%e A334184        |
%e A334184        1
%e A334184 Table begins:
%e A334184 1
%e A334184 1, 1
%e A334184 1, 1, 1
%e A334184 1, 1, 1
%e A334184 1, 1, 1, 1
%e A334184 1, 2, 1, 1
%e A334184 1, 1, 2, 1, 1
%e A334184 1, 1, 1, 1
%e A334184 1, 1, 2, 1, 1
%e A334184 1, 2, 1, 1, 1
%e A334184 1, 1, 2, 1, 1, 1
%e A334184 1, 2, 2, 1, 1
%e A334184 1, 1, 2, 2, 1, 1
%e A334184 1, 2, 2, 2, 1, 1
%e A334184 1, 2, 3, 2, 1, 1
%e A334184 1, 1, 1, 1, 1
%t A334184 Table[Length@ Union@ # & /@ Transpose@ # &@ If[n == 1, {{1}}, NestWhile[If[Length[#] == 0, Map[{n, #} &, # - # /FactorInteger[#][[All, 1]] ], Union[Join @@ Map[Function[{w, n}, Map[Append[w, If[n == 0, 0, n - n/#]] &, FactorInteger[n][[All, 1]] ]] @@ {#, Last@ #} &, #]]] &, n, If[ListQ[#], AllTrue[#, Last[#] > 1 &], # > 1] &]], {n, 22}] // Flatten (* _Michael De Vlieger_, Apr 18 2020 *)
%Y A334184 Cf. A001221, A073933, A332809, A333123, A334111, A334144, A334238.
%Y A334184 Cf. A000079, A019434.
%K A334184 nonn,tabf
%O A334184 1,15
%A A334184 _Peter Kagey_, Apr 17 2020