A372337 Rectangular array, read by descending antidiagonals: row n shows the numbers m > 1 in whose prime factorization p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k), all e(i) are <= 2 and the number of 0's in the multiset {e(i)} is n-1.
2, 4, 3, 6, 9, 5, 12, 10, 14, 7, 18, 15, 21, 22, 11, 30, 20, 25, 33, 26, 13, 36, 42, 28, 44, 39, 34, 17, 60, 45, 35, 49, 52, 51, 38, 19, 90, 50, 63, 55, 65, 68, 57, 46, 23, 150, 70, 66, 77, 91, 85, 76, 69, 58, 29, 180, 75, 98, 78, 102, 114, 95, 92, 87, 62
Offset: 1
Examples
28 = 2^2 * 3^0 * 5^0 * 7^1, so {e(i)} is {0,0,1,2}, so 28 is in row 3.
Corner:
2 4 6 12 18 30 36 60
3 9 10 15 20 42 45 50
5 14 21 25 28 35 63 66
7 22 33 44 49 55 77 78
11 26 39 52 65 91 102 117
13 34 51 68 85 114 119 153
17 38 57 76 95 133 138 171
19 46 69 92 115 161 174 207
23 58 87 116 145 186 203 261
Programs
-
Mathematica
exps := Map[#[[2]] &, Sort[Join[#, Complement[Map[{Prime[#], 0} &, Range[PrimePi[Last[#][[1]]]]], Map[{#[[1]], 0} &, #]]]] &[FactorInteger[#]]] &; m = Map[Transpose[#][[1]] &, GatherBy[Map[{#[[1]], Count[#[[2]], 0]} &, Select[Map[{#, exps[#]} &, Range[2, 7000]], Max[#[[2]]] <= 2 &]], #[[2]] &]]; z = 12; r = Table[Take[m[[n]], z], {n, 1, z}] Grid[r] (* array *) w[n_, k_] := r[[n]][[k]] Table[w[n - k + 1, k], {n, z}, {k, n, 1, -1}] // Flatten (* sequence *) (* Peter J. C. Moses, Mar 21 2024 *)