A370923 Rectangular array read by antidiagonals: row n shows the numbers m >=2 such that the maximum number of consecutive 0's in (e(1), e(2), ..., e(k)) is n-1, where p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k) is the prime factorization of m.

2, 4, 3, 6, 9, 5, 8, 10, 14, 7, 12, 15, 25, 22, 11, 16, 20, 28, 39, 26, 13, 18, 21, 33, 44, 51, 34, 17, 24, 27, 35, 49, 52, 57, 38, 19, 30, 40, 55, 77, 95, 68, 69, 46, 23, 32, 42, 56, 78, 102, 114, 76, 87, 58, 29, 36, 45, 65, 85, 104, 115, 138, 92, 93, 62

Offset: 1

Author

Clark Kimberling, Mar 18 2024

Keywords

• nonn
• tabl

Comments

Every positive integer >1 occurs exactly once.

Examples

			Corner:
   2   4    6    8   12   16   18   24   30
   3   9   10   15   20   21   27   40   42
   5  14   25   28   33   35   55   56   65
   7  22   39   44   49   77   78   85   88
  11  26   51   52   95  102  104  121  143
  13  34   57   68  114  115  136  169  171
  17  38   69   76  138  145  152  207  217
  19  46   87   92  155  174  184  259  261
  23  58   93  116  185  186  232  279  287
  29  62  111  124  205  222  248  301  333
  31  74  123  148  215  246  296  329  369
  37  82  129  164  235  258  328  371  387
22 = 2^1 * 3^0 * 5^0 * 7^0 * (11)^1, so (e(1),e(2),e(3),e(4),e(5)) = (1,0,0,0,1), so 22 is in row 4.
		

Crossrefs

Cf. A000040 (the primes, column 1), A002808 (union of all columns except the first), A055932 (row 1).

Programs

Mathematica

Map[Transpose[#][[1]] &, GatherBy[Map[{#, Max[Map[Length, DeleteCases[
Split[Map[IntegerQ, #/Prime[Range[PrimePi[FactorInteger[#][[-1, 1]]]]]] &[#]], {_, True, _}]] /. {} -> {0}]} &, Range[2, 400]], #[[2]] &]] // ColumnForm
(* Peter J. C. Moses, Mar 17 2024 *)