A372053 Irregular array read by rows: row n lists the factorizations of n into a product of nondecreasing integers >= 2.

2, 3, 2, 2, 4, 5, 2, 3, 6, 7, 2, 2, 2, 2, 4, 8, 3, 3, 9, 2, 5, 10, 11, 2, 2, 3, 2, 6, 3, 4, 12, 13, 2, 7, 14, 3, 5, 15, 2, 2, 2, 2, 2, 2, 4, 2, 8, 4, 4, 16, 17, 2, 3, 3, 2, 9, 3, 6, 18, 19, 2, 2, 5, 2, 10, 4, 5, 20, 3, 7, 21, 2, 11, 22, 23, 2, 2, 2, 3, 2, 2, 6, 2, 3, 4, 2, 12, 3, 8, 4, 6, 24, 5, 5, 25, 2, 13, 26, 3, 3, 3, 3, 9, 27, 2, 2, 7, 2, 14, 4, 7, 28, 29, 2, 3, 5, 2, 15, 3, 10, 5, 6, 30

Offset: 2

Author

N. J. A. Sloane, Apr 22 2024, following a suggestion from Scott R. Shannon

Keywords

• nonn
• tabf
• easy

Comments

Factorizations of n are ordered lexicographically. Compare A162247.

Examples

			The factorizations of the numbers 2 through 24 are:
  2;
  3;
  2, 2;  4;
  5;
  2, 3;  6;
  7;
  2, 2, 2;  2, 4;  8;
  3, 3;  9;
  2, 5;  10;
  11;
  2, 2, 3;  2, 6;  3, 4;  12;
  13;
  2, 7;  14;
  3, 5;  15;
  2, 2, 2, 2;  2, 2, 4;  2, 8;  4, 4;  16;
  17;
  2, 3, 3;  2, 9;  3, 6;  18;
  19;
  2, 2, 5;  2, 10;  4, 5;  20;
  3, 7;  21;
  2, 11;  22;
  23;
  2, 2, 2, 3;  2, 2, 6;  2, 3, 4;  2, 12;  3, 8;  4, 6;  24;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..19795 (rows n = 1..1000, flattened)

Crossrefs

Cf. A001055, A027746, A066637 (row lengths), A162247.

Programs

Mathematica

f[x_] := If[x <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[f[x/d], Min @@ # >= d &]], {d, Rest@ Divisors[x]}]]; Array[Flatten @* f, 29, 2] // Flatten (* Michael De Vlieger, Apr 22 2024 *)

Extensions

The DATA section is longer than usual in order to show the factorizations of 30.

Edited by Peter Munn, Feb 26 2025