OFFSET
0,2
COMMENTS
A permutation of the positive integers.
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.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..9295 (rows 0 <= n <= 25, flattened)
Michael De Vlieger, log-log plot of rows 0 <= n <= 30 of this sequence, highlighting 2^n in red and prime(n) in blue.
T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 11: {5} 56: {1,1,1,4}
2: {1} 64: {1,1,1,1,1,1} 45: {2,2,3}
4: {1,1} 48: {1,1,1,1,2} 50: {1,3,3}
3: {2} 36: {1,1,2,2} 42: {1,2,4}
8: {1,1,1} 40: {1,1,1,3} 44: {1,1,5}
6: {1,2} 27: {2,2,2} 35: {3,4}
5: {3} 30: {1,2,3} 33: {2,5}
16: {1,1,1,1} 28: {1,1,4} 26: {1,6}
12: {1,1,2} 25: {3,3} 17: {7}
9: {2,2} 21: {2,4} 256: {1,1,1,1,1,1,1,1}
10: {1,3} 22: {1,5} 192: {1,1,1,1,1,1,2}
7: {4} 13: {6} 144: {1,1,1,1,2,2}
32: {1,1,1,1,1} 128: {1,1,1,1,1,1,1} 160: {1,1,1,1,1,3}
24: {1,1,1,2} 96: {1,1,1,1,1,2} 108: {1,1,2,2,2}
18: {1,2,2} 72: {1,1,1,2,2} 120: {1,1,1,2,3}
20: {1,1,3} 80: {1,1,1,1,3} 112: {1,1,1,1,4}
15: {2,3} 54: {1,2,2,2} 81: {2,2,2,2}
14: {1,4} 60: {1,1,2,3} 90: {1,2,2,3}
The triangle begins:
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 25 21 22 13
128 96 72 80 54 60 56 45 50 42 44 35 33 26 17
MATHEMATICA
ralensort[f_, c_]:=If[Length[f]!=Length[c], Length[f]>Length[c], OrderedQ[{f, c}]];
Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n], ralensort], {n, 0, 8}]
CROSSREFS
Row lengths are A000041.
The constructive version is A228100.
Sorting by increasing length gives A334433.
The version with rows reversed is A334438.
Sum of prime indices is A056239.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
If the fine ordering is by Heinz number instead of lexicographic we get A333484.
KEYWORD
AUTHOR
Gus Wiseman, May 11 2020
EXTENSIONS
Name extended by Peter Luschny, Dec 23 2020
STATUS
approved