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%I #36 Apr 28 2020 05:58:12
%S 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 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 1,1,1,1,1,1,1,1,1,2,2,2,1,1,1,1,2,2,2
%N 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 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 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 Conjecture: rows are unimodal (increasing and then decreasing).
%C 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 Michael De Vlieger, <a href="/A334184/b334184.txt">Table of n, a(n) for n = 1..12386</a> (rows 1 <= n <= 1000, flattened)
%H 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 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 T(n,0) = T(n, A073933(n) - 2) = T(n, A073933(n) - 1) = 1.
%F T(n,1) = A001221(n) for n > 1.
%e 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 15
%e _/ \_
%e / \
%e 10 12
%e | \_ _/ |
%e | \ / |
%e 5 8 6
%e \_ | _/|
%e \_|_/ |
%e 4 3
%e | _/
%e |_/
%e 2
%e |
%e |
%e 1
%e Table begins:
%e 1
%e 1, 1
%e 1, 1, 1
%e 1, 1, 1
%e 1, 1, 1, 1
%e 1, 2, 1, 1
%e 1, 1, 2, 1, 1
%e 1, 1, 1, 1
%e 1, 1, 2, 1, 1
%e 1, 2, 1, 1, 1
%e 1, 1, 2, 1, 1, 1
%e 1, 2, 2, 1, 1
%e 1, 1, 2, 2, 1, 1
%e 1, 2, 2, 2, 1, 1
%e 1, 2, 3, 2, 1, 1
%e 1, 1, 1, 1, 1
%t 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 Cf. A001221, A073933, A332809, A333123, A334111, A334144, A334238.
%Y Cf. A000079, A019434.
%K nonn,tabf
%O 1,15
%A _Peter Kagey_, Apr 17 2020
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