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A334184
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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).
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9
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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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2
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OFFSET
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1,15
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COMMENTS
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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).
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).
Conjecture: rows are unimodal (increasing and then decreasing).
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
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LINKS
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Michael De Vlieger, Hasse diagrams showing rows n = {55, 63, 171, ...} that increase and decrease more than once.
Michael De Vlieger, Table of n, b(n) for n = 1..10000, encoding the running total of row n of this sequence as a binary number expressed decimally.
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FORMULA
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EXAMPLE
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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.
15
_/ \_
/ \
10 12
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5 8 6
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4 3
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2
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1
Table begins:
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, 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, 1, 1
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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