cp's OEIS Frontend

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.

%I A372337 #11 May 12 2024 11:26:45
%S A372337 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,
%T A372337 39,34,17,60,45,35,49,52,51,38,19,90,50,63,55,65,68,57,46,23,150,70,
%U A372337 66,77,91,85,76,69,58,29,180,75,98,78,102,114,95,92,87,62
%N 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.
%e A372337 28 = 2^2 * 3^0 * 5^0 * 7^1, so {e(i)} is {0,0,1,2}, so 28 is in row 3.
%e A372337 Corner:
%e A372337     2   4   6  12  18  30  36  60
%e A372337     3   9  10  15  20  42  45  50
%e A372337     5  14  21  25  28  35  63  66
%e A372337     7  22  33  44  49  55  77  78
%e A372337    11  26  39  52  65  91 102 117
%e A372337    13  34  51  68  85 114 119 153
%e A372337    17  38  57  76  95 133 138 171
%e A372337    19  46  69  92 115 161 174 207
%e A372337    23  58  87 116 145 186 203 261
%t A372337 exps := Map[#[[2]] &, Sort[Join[#, Complement[Map[{Prime[#], 0} &, Range[PrimePi[Last[#][[1]]]]], Map[{#[[1]], 0} &, #]]]] &[FactorInteger[#]]] &;
%t A372337 m = Map[Transpose[#][[1]] &, GatherBy[Map[{#[[1]], Count[#[[2]], 0]} &,
%t A372337 Select[Map[{#, exps[#]} &, Range[2, 7000]], Max[#[[2]]] <= 2 &]], #[[2]] &]];
%t A372337 z = 12; r = Table[Take[m[[n]], z], {n, 1, z}]
%t A372337 Grid[r]  (* array *)
%t A372337 w[n_, k_] := r[[n]][[k]]
%t A372337 Table[w[n - k + 1, k], {n, z}, {k, n, 1, -1}] // Flatten  (* sequence *)
%t A372337 (* _Peter J. C. Moses_, Mar 21 2024 *)
%Y A372337 Cf. A000040 (the primes, column 1), A371799.
%K A372337 nonn,tabl
%O A372337 1,1
%A A372337 _Clark Kimberling_, Apr 28 2024