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

1,1

Links

Table of n, a(n) for n=1..65.

Example

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

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 *)

Crossrefs

Cf. A000040 (the primes, column 1), A371799.

Sequence in context: A363473 A361748 A377093 * A371236 A246366 A271865

Adjacent sequences: A372334 A372335 A372336 * A372338 A372339 A372340

Keyword

nonn,tabl

Author

Clark Kimberling, Apr 28 2024

Status

approved