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%I #8 Apr 25 2024 13:38:03
%S 2,6,3,30,10,5,210,15,14,7,2310,42,21,22,11,30030,70,35,33,26,13,
%T 510510,105,66,55,39,34,17,9699690,330,110,77,65,51,38,19,223092870,
%U 462,154,78,91,85,57,46,23,6469693230,770,165,130,102,114,95,69,58,29
%N 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.
%e 15 = 2^0*3^1*51, so (e(1),e(2),e(3)) = (0,1,1), so 15 is in row 2
%e Corner:
%e 2 6 30 210 2310 30030 510510 9699690
%e 3 10 15 42 70 105 330 462
%e 5 14 21 35 66 110 154 165
%e 7 22 33 55 77 78 130 182
%e 11 26 39 65 91 102 143 170
%e 13 34 51 85 114 119 187 190
%e 17 38 57 95 133 138 209 230
%e 19 46 69 115 161 174 253 290
%e 23 58 87 145 186 203 310 319
%t exps := Map[#[[2]] &, Sort[Join[#, Complement[Map[{Prime[#], 0} &, Range[PrimePi[Last[#][[1]]]]], Map[{#[[1]], 0} &, #]]]] &[FactorInteger[#]]] &;
%t m = Map[Transpose[#][[1]] &, GatherBy[Map[{#[[1]], Count[#[[2]], 0]} &, Select[Map[{#, exps[#]} &, Range[2, 5000]], Max[#[[2]]] <= 1 &]], #[[2]] &]];
%t z = 12; row1 = Table[Apply[Times, Prime[Range[n]]], {n, 1, z}];
%t r = Join[{row1}, Table[Take[m[[n]], z], {n, 2, z}]];
%t Grid[r] (* array *)
%t w[n_, k_] := r[[n]][[k]]
%t Table[w[n - k + 1, k], {n, z}, {k, n, 1, -1}] // Flatten
%t (* sequence *)(* _Peter J. C. Moses_, Mar 21 2024 *)
%Y Cf. A000040 (the primes, column 1), A002110 (row 1), A005117 (increasing sequence of all terms of the array), A340316, A371801, A371802, A371803, A371804.
%K nonn,tabl
%O 1,1
%A _Clark Kimberling_, Apr 10 2024
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