A350604 Irregular triangle read by rows: row n lists the elements of the set S_n in increasing order, where S_1 = {1}, and S_{n+1} is the union of S_n, 2*S_n, and 3*S_n.

1, 1, 2, 3, 1, 2, 3, 4, 6, 9, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 54, 81, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 81, 108, 162, 243, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 144, 162, 216, 243, 324, 486, 729

Offset: 1

Author

N. J. A. Sloane, Jan 12 2022

Keywords

• nonn
• look
• tabf

Comments

S_n contains n*(n+1)/2 elements.

The rows converge to A003586.

Examples

			The first few sets S_n are:
  [1],
  [1, 2, 3],
  [1, 2, 3, 4, 6, 9],
  [1, 2, 3, 4, 6, 8, 9, 12, 18, 27],
  [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 54, 81],
  [1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 81, 108, 162, 243],
  ...
		

Links

• Alois P. Heinz, Rows n = 1..50, flattened

Crossrefs

Cf. A003586, A350603.

Programs

Maple

T:= proc(n) option remember; `if`(n=1, 1, sort(
     [map(k-> [k, 2*k, 3*k][], {T(n-1)})[]])[])
    end:
seq(T(n), n=1..7);  # Alois P. Heinz, Jan 12 2022

Mathematica

S[1] = {1};
S[n_] := S[n] = Union[S[n-1], 2*S[n-1], 3*S[n-1]];
Table[S[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Jun 02 2022 *)

Python

from itertools import chain, islice
def A350604_gen(): # generator of terms
    s = {1}
    while True:
        yield from sorted(s)
        s = set(chain.from_iterable((x,2*x,3*x) for x in s))
A350604_list = list(islice(A350604_gen(),30)) # Chai Wah Wu, Jan 12 2022