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

%I A372053 #24 Jun 21 2025 07:54:13
%S A372053 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,
%T A372053 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,
%U A372053 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
%N A372053 Irregular array read by rows: row n lists the factorizations of n into a product of nondecreasing integers >= 2.
%C A372053 Factorizations of n are ordered lexicographically. Compare A162247.
%H A372053 Michael De Vlieger, <a href="/A372053/b372053.txt">Table of n, a(n) for n = 2..19795</a> (rows n = 1..1000, flattened)
%e A372053 The factorizations of the numbers 2 through 24 are:
%e A372053   2;
%e A372053   3;
%e A372053   2, 2;  4;
%e A372053   5;
%e A372053   2, 3;  6;
%e A372053   7;
%e A372053   2, 2, 2;  2, 4;  8;
%e A372053   3, 3;  9;
%e A372053   2, 5;  10;
%e A372053   11;
%e A372053   2, 2, 3;  2, 6;  3, 4;  12;
%e A372053   13;
%e A372053   2, 7;  14;
%e A372053   3, 5;  15;
%e A372053   2, 2, 2, 2;  2, 2, 4;  2, 8;  4, 4;  16;
%e A372053   17;
%e A372053   2, 3, 3;  2, 9;  3, 6;  18;
%e A372053   19;
%e A372053   2, 2, 5;  2, 10;  4, 5;  20;
%e A372053   3, 7;  21;
%e A372053   2, 11;  22;
%e A372053   23;
%e A372053   2, 2, 2, 3;  2, 2, 6;  2, 3, 4;  2, 12;  3, 8;  4, 6;  24;
%t A372053 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 *)
%Y A372053 Cf. A001055, A027746, A066637 (row lengths), A162247.
%K A372053 nonn,tabf,easy
%O A372053 2,1
%A A372053 _N. J. A. Sloane_, Apr 22 2024, following a suggestion from _Scott R. Shannon_
%E A372053 The DATA section is longer than usual in order to show the factorizations of 30.
%E A372053 Edited by _Peter Munn_, Feb 26 2025