A333959 First occurrence of n in A334144.

1, 6, 15, 33, 65, 77, 154, 161, 217, 231, 455, 469, 483, 693, 957, 987, 1001, 1449, 1463, 2021, 2717, 2093, 2415, 2967, 3003, 4147, 3059, 4853, 4945, 4899, 6083, 8533, 4991, 7161, 9982, 8987, 9177, 10787, 10857, 10465, 10199, 12857, 14539, 20355, 18753, 20398

Offset: 1

Author

Michael De Vlieger, Peter Kagey, Antti Karttunen, Robert G. Wilson v, Apr 24 2020

Keywords

• nonn

Comments

Consider the mappings f(m) := m -> m - m/p across primes p | m.

Row m of A334184, read as a triangle T(m, k), lists the number of distinct values that proceed from the mapping after exactly k iterations.

A334144(m) is the largest value in row m of A334184.

The smallest term in this sequence that is not an index of a record in A334144 is a(22) = 2093.

From Robert G. Wilson v, Jun 14 2020: (Start)

All terms are nonprimes, but not necessarily squarefree. They are: 693, 1449, 91791, 13126113, 46334057, ..., .

Even terms: 6, 154, 9982, 20398, 29946, 812630, 1366666, 4263182, 17766658, 22866158, 34688186, 80633294, ..., .

Except for the initial even term, all even terms divided by 2 are also terms.

(End)

Examples

			1 is the first term since 1 is the empty product.
6 follows 1 since 2 <= m <= 5 have total order, thus the maximum number in A333184 is 1. For m = 6, the mapping f(m) has two distinct results {4, 3}, which generate chains {4, 2, 1} and {3, 2, 1}, respectively, with the last two terms in both chains coincident. Since the largest number of terms in an antichain is 2, a(2) = 6.
15 follows 6 since row 15 of A334184 = [1, 2, 3, 2, 1, 1] is the smallest m for which n = 3 appears.
Hasse diagrams of the 3 smallest terms, with brackets around the widest row.
[1]        6           15
          / \          /\
         /   \        /  \
        [4   3]     12  __10
         |  /       | \/   |
         | /        |_/\   |
         2         [8  _6  5]
         |          | /_|_/
         |          |// |
         1          4   3
                    |  /
                    |_/
                    2
                    |
                    |
                    1
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..871 (first 256 terms from Peter Kagey)
• Math StackExchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?, 2020.
• Michael De Vlieger, Hasse diagrams of a(n) for 2 <= n <= 37.

Crossrefs

Cf. A064097, A332809, A333123, A334144, A334184.

Programs

Mathematica

With[{s = Table[Max[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, 10^3}]}, TakeWhile[Array[FirstPosition[s, #][[1]] &, Max@ s], IntegerQ]]
f[n_] := Block[{lst = {{n}}}, While[lst[[-1]] != {1}, lst = Join[ lst, {Union[ Flatten[# - #/(First@# & /@ FactorInteger@#) & /@ lst[[-1]]] ]}]]; Max[Length@# & /@ lst]]; t[] := 0; k = 1; While[k < 21001, a = f@k; If[ t[a] == 0, t[a] = k]; k++]; t@# & /@ Range@ 46 (* _Robert G. Wilson v, Jun 14 2020 *)