A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.

1, 2, 4, 6, 9, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 2401, 32, 48, 72, 80, 108, 112, 120, 162

Offset: 0

Author

Robert Price, Mar 03 2020

Keywords

• nonn
• tabf

Comments

Positive integers not in T are: 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, ... .

Row n has exactly one squarefree member: primorial(n) = A002110(n).

Sorting all terms (except 1) gives A324521.

Examples

			Triangle T(n,k) begins:
  1;
  2;
  4,  6,  9;
  8, 12, 18, 20, 27, 30, 45, 50, 75, 125;
  ...
		

Links

• Robert Price, Table of n, a(n) for n = 0..8788

Crossrefs

Column k=1 gives A000079.

Last elements of rows give A307539.

Row lengths give A088218.

Row sums give A332967(n) = A124960(2n,n).

T(n,n) gives A101695(n) for n > 0.

Cf. A000720, A001222, A005117, A027748, A324521.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1], [seq(
      map(x-> x*ithprime(j), b(n-1, j))[], j=1..i)])
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Mar 03 2020

Mathematica

t = Table[Union[Apply[Times, Tuples[Prime[Range[n]], {n}], {1}]], {n, 0, 5}];
t // TableForm
Flatten[t]