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.

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

Links

Table of n, a(n) for n=1..55.

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

Adjacent sequences: A371796 A371797 A371798 * A371800 A371801 A371802

Keyword

nonn,tabl

Author

Clark Kimberling, Apr 10 2024

Status

approved