A360371 Triangle read by rows: lexicographically earliest sequence of distinct positive integers such that each column contains only multiples of the first number in that column. See example.

1, 2, 3, 4, 6, 5, 7, 9, 10, 8, 11, 12, 15, 16, 13, 14, 18, 20, 24, 26, 17, 19, 21, 25, 32, 39, 34, 22, 23, 27, 30, 40, 52, 51, 44, 28, 29, 33, 35, 48, 65, 68, 66, 56, 31, 36, 42, 45, 64, 78, 85, 88, 84, 62, 37, 38, 54, 50, 72, 91, 102, 110, 112, 93, 74, 41

Offset: 1

Author

Samuel Harkness, Mar 17 2023

Keywords

• nonn
• tabl

Comments

A permutation of the natural numbers.

Among the first number of columns, are there more primes or composites? Of the first 500 columns, 296 are prime, 203 are composite (first column begins with 1).

Examples

			The start of the sequence as a triangular array read by rows:
   1;
   2,  3;
   4,  6,  5;
   7,  9, 10,  8;
  11, 12, 15, 16, 13;
  14, 18, 20, 24, 26, 17;
  19, 21, 25, 32, 39, 34, 22;
  23, 27, 30, 40, 52, 51, 44, 28;
  ...
Note that each column contains only multiples of the first number in the column.
For a(17), note that we are in the second column, so a(17) must be a positive multiple of 3. No numbers can be repeated, and we see that {3, 6, 9, 12, 15} have already been used, and 18 is the smallest unused positive multiple of 3. Therefore, a(17) = 18.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10011 (first 141 rows flattened)
• Samuel Harkness, First 500000 terms
• Samuel Harkness, MATLAB program
• Rémy Sigrist, PARI program
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A194982, A361251 (inverse).

Programs

MATLAB

See Links section.

Maple

b:= proc() false end:
T:= proc(n, k) option remember; local j;
      if {n, k} = {1} then j:=1
    elif n=k then for j from T(n-1$2) while b(j) do od
    else for j from T(n-1, k) by T(k, k) while b(j) do od
      fi; b(j):=true; j
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 19 2023