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A129129
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An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.
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34
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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, 27, 36, 48, 64, 17, 26, 33, 44, 35, 42, 56, 50, 45, 60, 80, 54, 72, 96, 128, 19, 34, 39, 52, 55, 66, 88, 49, 70, 63, 84, 112, 75, 100, 90, 120, 160, 81, 108, 144, 192, 256
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OFFSET
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0,2
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COMMENTS
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The tree begins (at height n, n >= 0, nodes represent partitions of n)
0: 1
1: 2
2: 3 4
3: 5 6 8
4: 7 10 9 12 16
5: 11 14 15 20 18 24 32
...
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
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
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LINKS
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FORMULA
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(End)
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EXAMPLE
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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:
(1) pick an odd number (e.g., 135)
(2) calculate its prime factorization (135 = 5*3*3*3)
(3) note the least prime factor (LPF(135) = 3)
(4) note the index of the LPF (index(3) = 2)
(5) subtract one from the index (2-1 = 1)
(6) calculate the prime associated with the value in step five (prime(1) = 2)
(7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).
Partitions of 4 in graded reverse lexicographic order:
{4}: p_4 = 7;
{3,1}: p_3 * p_1 = 5 * 2 = 10;
{2,2}: p_2 * p_2 = 3^2 = 9;
{2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;
{1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)
The sequence together with the corresponding partitions begins:
1: () 24: (2,1,1,1) 35: (4,3)
2: (1) 32: (1,1,1,1,1) 42: (4,2,1)
3: (2) 13: (6) 56: (4,1,1,1)
4: (1,1) 22: (5,1) 50: (3,3,1)
5: (3) 21: (4,2) 45: (3,2,2)
6: (2,1) 28: (4,1,1) 60: (3,2,1,1)
8: (1,1,1) 25: (3,3) 80: (3,1,1,1,1)
7: (4) 30: (3,2,1) 54: (2,2,2,1)
10: (3,1) 40: (3,1,1,1) 72: (2,2,1,1,1)
9: (2,2) 27: (2,2,2) 96: (2,1,1,1,1,1)
12: (2,1,1) 36: (2,2,1,1) 128: (1,1,1,1,1,1,1)
16: (1,1,1,1) 48: (2,1,1,1,1) 19: (8)
11: (5) 64: (1,1,1,1,1,1) 34: (7,1)
14: (4,1) 17: (7) 39: (6,2)
15: (3,2) 26: (6,1) 52: (6,1,1)
20: (3,1,1) 33: (5,2) 55: (5,3)
18: (2,2,1) 44: (5,1,1) 66: (5,2,1)
(End)
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MAPLE
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b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),
b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> b(n$2)[]:
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MATHEMATICA
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Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* Michael De Vlieger, Aug 07 2018 *)
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[n_] := b[n, n];
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CROSSREFS
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Compositions under the same order are A066099.
The opposite version (sum/lex) is A334434.
The length-sensitive version (sum/length/revlex) is A334438.
The version for reversed (weakly increasing) partitions is A334436.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Sorting reversed partitions by Heinz number gives A112798.
Partitions in lexicographic order are A193073.
Sorting partitions by Heinz number gives A296150.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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