cp's OEIS Frontend

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.

%I A371799 #8 Apr 25 2024 13:38:03
%S A371799 2,6,3,30,10,5,210,15,14,7,2310,42,21,22,11,30030,70,35,33,26,13,
%T A371799 510510,105,66,55,39,34,17,9699690,330,110,77,65,51,38,19,223092870,
%U A371799 462,154,78,91,85,57,46,23,6469693230,770,165,130,102,114,95,69,58,29
%N 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.
%e A371799 15 = 2^0*3^1*51, so (e(1),e(2),e(3)) = (0,1,1), so 15 is in row 2
%e A371799 Corner:
%e A371799  2   6   30  210  2310  30030  510510  9699690
%e A371799  3   10  15   42    70    105     330      462
%e A371799  5   14  21   35    66    110     154      165
%e A371799  7   22  33   55    77     78     130      182
%e A371799 11   26  39   65    91    102     143      170
%e A371799 13   34  51   85   114    119     187      190
%e A371799 17   38  57   95   133    138     209      230
%e A371799 19   46  69  115   161    174     253      290
%e A371799 23   58  87  145   186    203     310      319
%t A371799 exps := Map[#[[2]] &, Sort[Join[#, Complement[Map[{Prime[#], 0} &, Range[PrimePi[Last[#][[1]]]]], Map[{#[[1]], 0} &, #]]]] &[FactorInteger[#]]] &;
%t A371799 m = Map[Transpose[#][[1]] &, GatherBy[Map[{#[[1]], Count[#[[2]], 0]} &,     Select[Map[{#, exps[#]} &, Range[2, 5000]], Max[#[[2]]] <= 1 &]], #[[2]] &]];
%t A371799 z = 12; row1 = Table[Apply[Times, Prime[Range[n]]], {n, 1, z}];
%t A371799 r = Join[{row1}, Table[Take[m[[n]], z], {n, 2, z}]];
%t A371799 Grid[r]  (* array *)
%t A371799 w[n_, k_] := r[[n]][[k]]
%t A371799 Table[w[n - k + 1, k], {n, z}, {k, n, 1, -1}] // Flatten
%t A371799 (* sequence *)(* _Peter J. C. Moses_, Mar 21 2024 *)
%Y A371799 Cf. A000040 (the primes, column 1), A002110 (row 1), A005117 (increasing sequence of all terms of the array), A340316, A371801, A371802, A371803, A371804.
%K A371799 nonn,tabl
%O A371799 1,1
%A A371799 _Clark Kimberling_, Apr 10 2024