This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A129129 #55 Sep 22 2023 07:55:56
%S A129129 1,2,3,4,5,6,8,7,10,9,12,16,11,14,15,20,18,24,32,13,22,21,28,25,30,40,
%T A129129 27,36,48,64,17,26,33,44,35,42,56,50,45,60,80,54,72,96,128,19,34,39,
%U A129129 52,55,66,88,49,70,63,84,112,75,100,90,120,160,81,108,144,192,256
%N A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.
%C A129129 The tree begins (at height n, n >= 0, nodes represent partitions of n)
%C A129129 0: 1
%C A129129 1: 2
%C A129129 2: 3 4
%C A129129 3: 5 6 8
%C A129129 4: 7 10 9 12 16
%C A129129 5: 11 14 15 20 18 24 32
%C A129129 ...
%C A129129 and hence differs from A114622.
%C A129129 Ordering [graded reverse lexicographic order] of partitions (positive integer representation) of nonnegative integers, where part of size i [as summand] is mapped to i-th prime [as multiplicand], where the empty partition for 0 yields the empty product, i.e., 1. Permutation of positive integers, since bijection [1-1 and onto map] between the set of all partitions of nonnegative integers and positive integers. - _Daniel Forgues_, Aug 07 2018
%C A129129 These are all Heinz numbers of integer partitions in graded reverse-lexicographic order, where The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is the so-called "Mathematica" order (sum/revlex) of partitions (A080577). Partitions in lexicographic order (sum/lex) are A193073, with Heinz numbers A334434. - _Gus Wiseman_, May 19 2020
%H A129129 Michael De Vlieger, <a href="/A129129/b129129.txt">Table of n, a(n) for n = 0..11731</a> (rows 0 <= n <= 26).
%H A129129 OEIS Wiki, <a href="/wiki/Partitions#Orderings_of_partitions">Partitions#Orderings of partitions</a>.
%H A129129 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>.
%F A129129 From _Gus Wiseman_, May 19 2020: (Start)
%F A129129 A001222(a(n)) = A238966(n).
%F A129129 A001221(a(n)) = A115623(n).
%F A129129 A056239(a(n)) = A036042(n).
%F A129129 A061395(a(n)) = A331581(n).
%F A129129 (End)
%e A129129 The array is a tree structure as described by A128628. If a node value has only one branch the value is twice that of its parent node. If it has two branches one is twice that of its parent node but the other is defined as indicated below:
%e A129129 (1) pick an odd number (e.g., 135)
%e A129129 (2) calculate its prime factorization (135 = 5*3*3*3)
%e A129129 (3) note the least prime factor (LPF(135) = 3)
%e A129129 (4) note the index of the LPF (index(3) = 2)
%e A129129 (5) subtract one from the index (2-1 = 1)
%e A129129 (6) calculate the prime associated with the value in step five (prime(1) = 2)
%e A129129 (7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).
%e A129129 From _Daniel Forgues_, Aug 07 2018: (Start)
%e A129129 Partitions of 4 in graded reverse lexicographic order:
%e A129129 {4}: p_4 = 7;
%e A129129 {3,1}: p_3 * p_1 = 5 * 2 = 10;
%e A129129 {2,2}: p_2 * p_2 = 3^2 = 9;
%e A129129 {2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;
%e A129129 {1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)
%e A129129 From _Gus Wiseman_, May 19 2020: (Start)
%e A129129 The sequence together with the corresponding partitions begins:
%e A129129 1: () 24: (2,1,1,1) 35: (4,3)
%e A129129 2: (1) 32: (1,1,1,1,1) 42: (4,2,1)
%e A129129 3: (2) 13: (6) 56: (4,1,1,1)
%e A129129 4: (1,1) 22: (5,1) 50: (3,3,1)
%e A129129 5: (3) 21: (4,2) 45: (3,2,2)
%e A129129 6: (2,1) 28: (4,1,1) 60: (3,2,1,1)
%e A129129 8: (1,1,1) 25: (3,3) 80: (3,1,1,1,1)
%e A129129 7: (4) 30: (3,2,1) 54: (2,2,2,1)
%e A129129 10: (3,1) 40: (3,1,1,1) 72: (2,2,1,1,1)
%e A129129 9: (2,2) 27: (2,2,2) 96: (2,1,1,1,1,1)
%e A129129 12: (2,1,1) 36: (2,2,1,1) 128: (1,1,1,1,1,1,1)
%e A129129 16: (1,1,1,1) 48: (2,1,1,1,1) 19: (8)
%e A129129 11: (5) 64: (1,1,1,1,1,1) 34: (7,1)
%e A129129 14: (4,1) 17: (7) 39: (6,2)
%e A129129 15: (3,2) 26: (6,1) 52: (6,1,1)
%e A129129 20: (3,1,1) 33: (5,2) 55: (5,3)
%e A129129 18: (2,2,1) 44: (5,1,1) 66: (5,2,1)
%e A129129 (End)
%p A129129 b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),
%p A129129 b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A129129 T:= n-> b(n$2)[]:
%p A129129 seq(T(n), n=0..10); # _Alois P. Heinz_, Feb 14 2020
%t A129129 Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* _Michael De Vlieger_, Aug 07 2018 *)
%t A129129 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {2^n}, Join[(# Prime[i]&) /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
%t A129129 T[n_] := b[n, n];
%t A129129 T /@ Range[0, 10] // Flatten (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)
%Y A129129 Cf. A080577 (the partitions), A252461, A114622, A128628, A215366 (sorted rows).
%Y A129129 Row lengths are A000041.
%Y A129129 Compositions under the same order are A066099.
%Y A129129 The opposite version (sum/lex) is A334434.
%Y A129129 The length-sensitive version (sum/length/revlex) is A334438.
%Y A129129 The version for reversed (weakly increasing) partitions is A334436.
%Y A129129 Lexicographically ordered reversed partitions are A026791.
%Y A129129 Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
%Y A129129 Sum of prime indices is A056239.
%Y A129129 Sorting reversed partitions by Heinz number gives A112798.
%Y A129129 Partitions in lexicographic order are A193073.
%Y A129129 Sorting partitions by Heinz number gives A296150.
%Y A129129 Cf. A036037, A185974, A211992, A334301, A334433, A334435, A334437, A334439.
%K A129129 nonn,tabf
%O A129129 0,2
%A A129129 _Alford Arnold_, Mar 31 2007