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A333485 Heinz numbers of all integer partitions sorted first by sum, then by decreasing length, and finally lexicographically. A code for the Fenner-Loizou tree A228100.

%I A333485 #20 Feb 09 2021 18:34:45
%S A333485 1,2,4,3,8,6,5,16,12,9,10,7,32,24,18,20,15,14,11,64,48,36,40,27,30,28,
%T A333485 25,21,22,13,128,96,72,80,54,60,56,45,50,42,44,35,33,26,17,256,192,
%U A333485 144,160,108,120,112,81,90,100,84,88,75,63,70,66,52,49,55,39,34,19
%N A333485 Heinz numbers of all integer partitions sorted first by sum, then by decreasing length, and finally lexicographically. A code for the Fenner-Loizou tree A228100.
%C A333485 A permutation of the positive integers.
%C A333485 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), which gives a bijective correspondence between positive integers and integer partitions.
%C A333485 As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.
%H A333485 Michael De Vlieger, <a href="/A333485/b333485.txt">Table of n, a(n) for n = 0..9295</a> (rows 0 <= n <= 25, flattened)
%H A333485 Michael De Vlieger, <a href="/A333485/a333485.png">log-log plot of rows 0 <= n <= 30 of this sequence</a>, highlighting 2^n in red and prime(n) in blue.
%H A333485 T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
%H A333485 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A333485 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A333485 A001221(a(n)) = A115623(n).
%F A333485 A001222(a(n - 1)) = A331581(n).
%F A333485 A061395(a(n > 1)) = A128628(n).
%e A333485 The sequence of terms together with their prime indices begins:
%e A333485     1: {}              11: {5}                 56: {1,1,1,4}
%e A333485     2: {1}             64: {1,1,1,1,1,1}       45: {2,2,3}
%e A333485     4: {1,1}           48: {1,1,1,1,2}         50: {1,3,3}
%e A333485     3: {2}             36: {1,1,2,2}           42: {1,2,4}
%e A333485     8: {1,1,1}         40: {1,1,1,3}           44: {1,1,5}
%e A333485     6: {1,2}           27: {2,2,2}             35: {3,4}
%e A333485     5: {3}             30: {1,2,3}             33: {2,5}
%e A333485    16: {1,1,1,1}       28: {1,1,4}             26: {1,6}
%e A333485    12: {1,1,2}         25: {3,3}               17: {7}
%e A333485     9: {2,2}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
%e A333485    10: {1,3}           22: {1,5}              192: {1,1,1,1,1,1,2}
%e A333485     7: {4}             13: {6}                144: {1,1,1,1,2,2}
%e A333485    32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    160: {1,1,1,1,1,3}
%e A333485    24: {1,1,1,2}       96: {1,1,1,1,1,2}      108: {1,1,2,2,2}
%e A333485    18: {1,2,2}         72: {1,1,1,2,2}        120: {1,1,1,2,3}
%e A333485    20: {1,1,3}         80: {1,1,1,1,3}        112: {1,1,1,1,4}
%e A333485    15: {2,3}           54: {1,2,2,2}           81: {2,2,2,2}
%e A333485    14: {1,4}           60: {1,1,2,3}           90: {1,2,2,3}
%e A333485 The triangle begins:
%e A333485     1
%e A333485     2
%e A333485     4   3
%e A333485     8   6   5
%e A333485    16  12   9  10   7
%e A333485    32  24  18  20  15  14  11
%e A333485    64  48  36  40  27  30  28  25  21  22  13
%e A333485   128  96  72  80  54  60  56  45  50  42  44  35  33  26  17
%t A333485 ralensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]>Length[c],OrderedQ[{f,c}]];
%t A333485 Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n],ralensort],{n,0,8}]
%Y A333485 Row lengths are A000041.
%Y A333485 The constructive version is A228100.
%Y A333485 Sorting by increasing length gives A334433.
%Y A333485 The version with rows reversed is A334438.
%Y A333485 Sum of prime indices is A056239.
%Y A333485 Reverse-lexicographically ordered partitions are A080577.
%Y A333485 Sorting reversed partitions by Heinz number gives A112798.
%Y A333485 Lexicographically ordered partitions are A193073.
%Y A333485 Graded Heinz numbers are A215366.
%Y A333485 Sorting partitions by Heinz number gives A296150.
%Y A333485 If the fine ordering is by Heinz number instead of lexicographic we get A333484.
%Y A333485 Cf. A000041, A026791, A036036, A036037, A036043, A185974, A211992, A228351, A296773, A333483, A334301, A334439.
%K A333485 nonn,look,tabf
%O A333485 0,2
%A A333485 _Gus Wiseman_, May 11 2020
%E A333485 Name extended by _Peter Luschny_, Dec 23 2020