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

%I A333483 #8 May 28 2020 05:00:12
%S A333483 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,
%T A333483 36,40,48,64,17,26,33,35,42,44,45,50,54,56,60,72,80,96,128,19,34,39,
%U A333483 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
%N A333483 Sort all positive integers, first by sum of prime indices (A056239), then by number of prime indices (A001222).
%C A333483 A refinement of A215366, from which it first differs at a(49) = 55, A215366(49) = 52.
%C A333483 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 A333483 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A333483 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A333483 Triangle begins:
%e A333483    1
%e A333483    2
%e A333483    3   4
%e A333483    5   6   8
%e A333483    7   9  10  12  16
%e A333483   11  14  15  18  20  24  32
%e A333483   13  21  22  25  27  28  30  36  40  48  64
%e A333483   17  26  33  35  42  44  45  50  54  56  60  72  80  96 128
%t A333483 Join@@@Table[Sort[Times@@Prime/@#&/@IntegerPartitions[n,{k}]],{n,0,8},{k,0,n}]
%Y A333483 Row lengths are A000041.
%Y A333483 Ignoring length gives A215366 (graded Heinz numbers).
%Y A333483 Sorting by decreasing length gives A333484.
%Y A333483 Finally sorting lexicographically by prime indices gives A185974.
%Y A333483 Finally sorting colexicographically by prime indices gives A334433.
%Y A333483 Finally sorting reverse-lexicographically by prime indices gives A334435.
%Y A333483 Finally sorting reverse-colexicographically by prime indices gives A334438.
%Y A333483 Number of prime indices is A001222.
%Y A333483 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A333483 Partitions in (sum/length/colex) order are A036037.
%Y A333483 Sum of prime indices is A056239.
%Y A333483 Sorting reversed partitions by Heinz number gives A112798.
%Y A333483 Sorting partitions by Heinz number gives A296150.
%Y A333483 Cf. A026791, A124734, A129129, A193073, A211992, A228100, A333219, A334301, A334434, A334439, A334441, A334442.
%K A333483 nonn,tabf
%O A333483 0,2
%A A333483 _Gus Wiseman_, May 10 2020