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.

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, 256, 192, 144, 160, 108, 120, 112, 81, 90, 100, 84, 88, 75, 63, 70, 66, 52, 49, 55, 39, 34, 19

Offset: 0

Author

Gus Wiseman, May 11 2020

Keywords

• nonn
• look
• tabf

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.

Examples

			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
		

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

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.

Cf. A000041, A026791, A036036, A036037, A036043, A185974, A211992, A228351, A296773, A333483, A334301, A334439.

Programs

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}]

Formula

A001221(a(n)) = A115623(n).

A001222(a(n - 1)) = A331581(n).

A061395(a(n > 1)) = A128628(n).

Extensions

Name extended by Peter Luschny, Dec 23 2020