A333483 Sort all positive integers, first by sum of prime indices (A056239), then by number of prime indices (A001222).

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128, 19, 34, 39, 49, 55, 52, 63, 66, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 256, 23, 38, 51, 65, 77, 68, 78, 98, 99, 105, 110, 125, 104, 126, 132, 135, 140, 150, 162, 168, 176, 180, 200, 216, 224, 240, 288, 320, 384, 512

Offset: 0

Author

Gus Wiseman, May 10 2020

Keywords

• nonn
• tabf

Comments

A refinement of A215366, from which it first differs at a(49) = 55, A215366(49) = 52.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  14  15  18  20  24  32
  13  21  22  25  27  28  30  36  40  48  64
  17  26  33  35  42  44  45  50  54  56  60  72  80  96 128
		

Links

• OEIS Wiki, Orderings of partitions
• Wikiversity, Lexicographic and colexicographic order

Crossrefs

Row lengths are A000041.

Ignoring length gives A215366 (graded Heinz numbers).

Sorting by decreasing length gives A333484.

Finally sorting lexicographically by prime indices gives A185974.

Finally sorting colexicographically by prime indices gives A334433.

Finally sorting reverse-lexicographically by prime indices gives A334435.

Finally sorting reverse-colexicographically by prime indices gives A334438.

Number of prime indices is A001222.

Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.

Partitions in (sum/length/colex) order are A036037.

Sum of prime indices is A056239.

Sorting reversed partitions by Heinz number gives A112798.

Sorting partitions by Heinz number gives A296150.

Cf. A026791, A124734, A129129, A193073, A211992, A228100, A333219, A334301, A334434, A334439, A334441, A334442.

Programs

Mathematica

Join@@@Table[Sort[Times@@Prime/@#&/@IntegerPartitions[n,{k}]],{n,0,8},{k,0,n}]