A324939 Triangle T(n,k) read by rows in which n-th row lists in increasing order all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i); n>=0, 1<=k<=A011782(n).

1, 2, 4, 6, 8, 12, 18, 30, 16, 24, 36, 54, 60, 90, 150, 210, 32, 48, 72, 108, 120, 162, 180, 270, 300, 420, 450, 630, 750, 1050, 1470, 2310, 64, 96, 144, 216, 240, 324, 360, 486, 540, 600, 810, 840, 900, 1260, 1350, 1500, 1890, 2100, 2250, 2940, 3150, 3750, 4410, 4620, 5250, 6930, 7350, 10290, 11550, 16170, 25410, 30030

Offset: 0

Author

Alois P. Heinz, Sep 04 2019

Keywords

• nonn
• tabf

Comments

All terms sorted give A055932.

All terms first sorted by number of factors give A057335.

Examples

			Triangle T(n,k) begins:
   1;
   2;
   4,  6;
   8, 12, 18,  30;
  16, 24, 36,  54,  60,  90, 150, 210;
  32, 48, 72, 108, 120, 162, 180, 270, 300, 420, 450, 630, 750, 1050, 1470, 2310;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..16, flattened
• Wikipedia, Gödel's encoding

Crossrefs

Column k=1 gives A000079.

Last elements of rows give A002110.

Row sums give A325054.

Row lengths give A011782.

Cf. A000040, A055932, A057335, A087443, A215366.

Programs

Maple

b:= n-> `if`(n=0, [[]], [seq(map(x-> [j, x[]], b(n-j))[], j=1..n)]):
T:= n-> sort(map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n)))[]:
seq(T(n), n=0..7);