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A333484 Sort all positive integers, first by sum of prime indices (A056239), then by decreasing number of prime indices (A001222).

%I A333484 #6 May 10 2020 19:27:44
%S A333484 1,2,4,3,8,6,5,16,12,9,10,7,32,24,18,20,14,15,11,64,48,36,40,27,28,30,
%T A333484 21,22,25,13,128,96,72,80,54,56,60,42,44,45,50,26,33,35,17,256,192,
%U A333484 144,160,108,112,120,81,84,88,90,100,52,63,66,70,75,34,39,49,55,19
%N A333484 Sort all positive integers, first by sum of prime indices (A056239), then by decreasing number of prime indices (A001222).
%C A333484 A refinement of A215366.
%C A333484 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.
%H A333484 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A333484 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A333484 Triangle begins:
%e A333484     1
%e A333484     2
%e A333484     4   3
%e A333484     8   6   5
%e A333484    16  12   9  10   7
%e A333484    32  24  18  20  14  15  11
%e A333484    64  48  36  40  27  28  30  21  22  25  13
%e A333484   128  96  72  80  54  56  60  42  44  45  50  26  33  35  17
%t A333484 Join@@@Table[Sort[Times@@Prime/@#&/@IntegerPartitions[n,{k}]],{n,0,8},{k,n,0,-1}]
%Y A333484 Row lengths are A000041.
%Y A333484 Ignoring length gives A215366 (graded Heinz numbers).
%Y A333484 Sorting by increasing length gives A333483.
%Y A333484 Number of prime indices is A001222.
%Y A333484 Lexicographically ordered reversed partitions are A026791.
%Y A333484 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A333484 Partitions in (sum/length/colex) order are A036037.
%Y A333484 Sum of prime indices is A056239.
%Y A333484 Reverse-lexicographically ordered partitions are A080577.
%Y A333484 Sorting reversed partitions by Heinz number gives A112798.
%Y A333484 Lexicographically ordered partitions are A193073.
%Y A333484 Sorting partitions by Heinz number gives A296150.
%Y A333484 Cf. A124734, A129129, A211992, A228100, A333219, A334301, A334433, A334434, A334439, A334441, A334442.
%K A333484 nonn,tabf
%O A333484 0,2
%A A333484 _Gus Wiseman_, May 10 2020