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).

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

Offset: 1

Author

Peter Kagey, Apr 17 2020

Keywords

• nonn
• tabf

Comments

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

Examples

			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
  | \_   _/ |
  |   \ /   |
  5    8    6
   \_  |  _/|
     \_|_/  |
       4    3
       |  _/
       |_/
       2
       |
       |
       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
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..12386 (rows 1 <= n <= 1000, flattened)
• 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.

Crossrefs

Cf. A001221, A073933, A332809, A333123, A334111, A334144, A334238.

Cf. A000079, A019434.

Programs

Mathematica

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 *)

Formula

T(n,0) = T(n, A073933(n) - 2) = T(n, A073933(n) - 1) = 1.

T(n,1) = A001221(n) for n > 1.