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A371799
Rectangular array, read by downward 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 <= 1 and the number of 0' s in {e(i)} is n-1.
1
2, 6, 3, 30, 10, 5, 210, 15, 14, 7, 2310, 42, 21, 22, 11, 30030, 70, 35, 33, 26, 13, 510510, 105, 66, 55, 39, 34, 17, 9699690, 330, 110, 77, 65, 51, 38, 19, 223092870, 462, 154, 78, 91, 85, 57, 46, 23, 6469693230, 770, 165, 130, 102, 114, 95, 69, 58, 29
OFFSET
1,1
EXAMPLE
15 = 2^0*3^1*51, so (e(1),e(2),e(3)) = (0,1,1), so 15 is in row 2
Corner:
2 6 30 210 2310 30030 510510 9699690
3 10 15 42 70 105 330 462
5 14 21 35 66 110 154 165
7 22 33 55 77 78 130 182
11 26 39 65 91 102 143 170
13 34 51 85 114 119 187 190
17 38 57 95 133 138 209 230
19 46 69 115 161 174 253 290
23 58 87 145 186 203 310 319
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, 5000]], Max[#[[2]]] <= 1 &]], #[[2]] &]];
z = 12; row1 = Table[Apply[Times, Prime[Range[n]]], {n, 1, z}];
r = Join[{row1}, Table[Take[m[[n]], z], {n, 2, 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), A002110 (row 1), A005117 (increasing sequence of all terms of the array), A340316, A371801, A371802, A371803, A371804.
Sequence in context: A303761 A283478 A125666 * A307540 A304087 A284003
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Apr 10 2024
STATUS
approved