A371164 Rectangular array, read by antidiagonals: row k shows all numbers whose prime factorization p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k) has e(i) > 0 for i=1..k.

2, 4, 6, 8, 12, 30, 16, 18, 60, 210, 32, 24, 90, 420, 2310, 64, 36, 120, 630, 4620, 30030, 128, 48, 150, 840, 6930, 60060, 510510, 256, 54, 180, 1050, 9240, 90090, 1021020, 9699690, 512, 72, 240, 1260, 11550, 120120, 1531530, 19399380, 223092870, 1024, 96

Offset

1,1

Links

Table of n, a(n) for n=1..47.

Example

Corner:
      2      4      8      16      32     64
      6     12     18      24      36     48
     30     60     90     120     150    180
    210    420    630     840    1050   1260
   2310   4620   6930    9240   11550  13860
  30030  60060  90090  120120  150150 180180
90 = 2^1 * 3^2 * 5^1, so 90 is in row 3.

Mathematica

rows = 10; t = (Table[n = 120000/Exp[row] Apply[Times, Table[Prime[k], {k, row}]];
eq = Table[Prime[i]^k[i], {i, row}];
Sort[Flatten[Table[Apply[Times, eq], ##] & @@Table[{k[i],
   Log[Prime[i], n/Apply[Times, Take[eq, i - 1]]]}, {i, row}]]], {row, rows}]);
Table[t[[n]][[k]], {n, 1, rows}, {k, 1, rows}] // TableForm  (* array *)
w[n_, k_] := t[[n]][[k]];
Table[w[n - k + 1, k], {n, rows}, {k, n, 1, -1}] // Flatten   (* sequence *)
(* Peter J. C. Moses, Mar 23 2024 *)

Crossrefs

Cf. A000040, A055932 (union of all rows, preceded by 1).

Sequence in context: A131885 A173941 A194406 * A087443 A059901 A036035

Adjacent sequences: A371161 A371162 A371163 * A371165 A371166 A371167

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 26 2024

Status

approved