A036037 -id:A036037 - cp's OEIS frontend

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 3, 2, 5, 3, 2, 1, 5, 1, 4, 2, 6, 4, 2, 1, 6, 1, 5, 2, 4, 3, 7, 5, 2, 1, 4, 3, 1, 7, 1, 6, 2, 5, 3, 8, 6, 2, 1, 5, 3, 1, 8, 1, 4, 3, 2, 7, 2, 6, 3, 5, 4, 9, 4, 3, 2, 1, 7, 2, 1, 6, 3, 1, 5, 4, 1, 9, 1, 5, 3, 2, 8, 2, 7, 3, 6, 4, 10

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 2, 3, 1, 4, 5, 1, 2, 3, 2, 4, 1, 5, 6, 1, 2, 4, 3, 4, 2, 5, 1, 6, 7, 1, 3, 4, 1, 2, 5, 3, 5, 2, 6, 1, 7, 8, 2, 3, 4, 1, 3, 5, 4, 5, 1, 2, 6, 3, 6, 2, 7, 1, 8, 9, 1, 2, 3, 4, 2, 3, 5, 1, 4, 5, 1, 3, 6, 4, 6, 1, 2, 7, 3, 7, 2, 8, 1, 9, 10

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (12)(3)
  4: (13)(4)
  5: (23)(14)(5)
  6: (123)(24)(15)(6)
  7: (124)(34)(25)(16)(7)
  8: (134)(125)(35)(26)(17)(8)
  9: (234)(135)(45)(126)(36)(27)(18)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
The non-strict version is A080576.
Taking lex instead of colex gives A246688 (non-reversed: A344086).
The non-reversed version is A344087.
Taking revlex instead of colex gives A344089 (non-reversed: A118457).
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A181087 Partitions of n in the order of increasing smallest numbers of prime signatures.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 4, 1, 3, 1, 1, 1, 5, 2, 2, 1, 4, 1, 1, 2, 6, 2, 3, 1, 5, 1, 1, 3, 7, 2, 4, 1, 2, 2, 1, 6, 1, 1, 1, 1, 3, 3, 1, 1, 4, 8, 2, 5, 1, 2, 3, 1, 7, 1, 1, 1, 2, 3, 4, 1, 1, 5, 9, 2, 6, 1, 2, 4, 1, 8, 1, 1, 1, 3, 3, 5, 2, 2, 2, 1, 1, 6, 10, 1, 3, 3, 2, 7, 1, 1, 2, 2, 4, 4, 1, 2, 5, 1, 9, 1, 1, 1, 4, 3, 6, 2, 2, 3, 1, 1, 7, 11, 1, 3, 4, 2, 8, 1, 1

Offset: 1

Alois P. Heinz, Jan 23 2011

The parts of each partition are listed in increasing order.

			Smallest number with prime signature [1,1,1] is 2^1*3^1*5^1 = 30, the smallest number for [4] is 2^4 = 16, and thus [4] < [1,1,1] in this order.
First partitions in the order of increasing smallest numbers of prime signatures are: [1], [2], [1,1], [3], [1,2], [4], [1,3], [1,1,1], [5], [2,2], [1,4], [1,1,2], [6], [2,3], [1,5], [1,1,3], [7], [2,4], ...
Smallest numbers with these prime signatures are:  2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, ... A025487

Alois P. Heinz, Table of n, a(n) for n = 1..18132

Cf. A036036, A036037, A080576, A025487, A095904.

DeleteDuplicates[Map[Sort[Map[Last, FactorInteger[#]]] &, Range[1000]]] // Grid (* Geoffrey Critzer, Nov 27 2015 *)

def A181087_build(w):
    seen = set()
    a = []
    for n in PositiveIntegers():
        psig = tuple(sorted(m for p,m in factor(n)))
        if psig not in seen:
            a.extend(psig)
            seen.add(psig)
            if len(a) >= w: return a  # D. S. McNeil, Jan 23 2011

A344091 Flattened tetrangle of all finite multisets of positive integers sorted first by sum, then by length, then colexicographically.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 5, 2, 3, 1, 4, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 3, 3, 2, 4, 1, 5, 2, 2, 2, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 12 2021

First differs from A334302 for partitions of 9.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)(11)
  3: (3)(12)(111)
  4: (4)(22)(13)(112)(1111)
  5: (5)(23)(14)(122)(113)(1112)(11111)
  6: (6)(33)(24)(15)(222)(123)(114)(1122)(1113)(11112)(111111)

Wikiversity, Lexicographic and colexicographic order

The version for lex instead of colex is A036036.
Starting with reversed partitions gives A036037.
Ignoring length gives A211992 (reversed: A080576).
Same as A334301 with partitions reversed.
The version for revlex instead of colex is A334302.
The Heinz numbers of these partitions are A334433.
The strict case is A344089.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A193073 reads off lexicographically ordered partitions.
A296150 reads off partitions by Heinz number.
Cf. A000041, A026793, A124734, A185974, A228531, A246688, A296774, A334433, A334435, A334438, A334439, A334441, A334442, A344086, A344090.

Table[Reverse/@Sort[IntegerPartitions[n]],{n,0,9}]

A182937 Triangle in which n-th row lists all integer partitions of n, in order of traversing the periphery of the Fenner-Loizou tree in the clockwise sense.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 3, 1, 4, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 4, 1, 5, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 5, 1, 6, 4, 2, 3, 2, 1, 3, 3, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Peter Luschny, Jan 21 2011

If the Fenner-Loizou tree is traversed in the counterclockwise sense (preorder traversal) the integer partitions are in lexicographic order.

			First five rows are:
[[1]]
[[1, 1], [2]]
[[1, 1, 1], [2, 1], [3]]
[[1, 1, 1, 1], [2, 1, 1], [3, 1], [4], [2, 2]]
[[1, 1, 1, 1, 1], [2, 1, 1, 1], [3, 1, 1], [4, 1], [5], [3, 2], [2, 2,1]]

T. I. Fenner and G. Loizou, Comp. J. 23 (1980), 332-337.
D. E. Knuth, TAOCP 4 (2005), fasc. 3, 7.2.1.4, exercise 10.

Peter Luschny, Integer Partition Trees, OEIS wiki.

See A036036 for the Hindenburg (graded reflected colexicographic) ordering.
See A036037 for the graded colexicographic ordering.
See A080576 for the Maple (graded reflected lexicographic) ordering.
See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A193073 for the graded lexicographic ordering.
See A228100 for the Fenner-Loizou (binary tree) ordering.

A296010 Sum of the squares of the number of parts in all partitions of n.

Original entry on oeis.org

0, 1, 5, 14, 34, 68, 133, 232, 402, 652, 1048, 1609, 2465, 3640, 5358, 7694, 10993, 15399, 21498, 29520, 40394, 54572, 73425, 97756, 129710, 170525, 223428, 290552, 376551, 484819, 622317, 794167, 1010515, 1279376, 1615126, 2029948, 2544600, 3176856, 3956277

Offset: 0

Matthew C. Russell, Dec 02 2017

nonn

			For n=4, the 5 partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. These have 1, 2, 2, 3, and 4 parts, respectively. The sum of the squares is 1+4+4+9+16=34.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms 0..115 from Robert G. Wilson v)

Cf. A006128, A008284, A036037.

K:=[]:
for n from 0 to 20 do
co:=0:
for L in combinat[partition](n) do
co:=co+nops(L)^2:
od:
K:=[op(K),co]:
od:
K;
# second Maple program:
b:= proc(n, i, c) option remember; `if`(n=0 or i=1,
      (n+c)^2, `if`(i>n, 0, b(n-i, i, c+1))+b(n, i-1, c))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 02 2017

f[n_] := Sum[i^2 (Length@ IntegerPartitions[n, {i}]), {i, n}]; Array[f, 34, 0] (* Robert G. Wilson v, Dec 02 2017 *)
b[n_, i_, c_] := b[n, i, c] = If[n == 0 || i == 1,
     (n + c)^2, If[i > n, 0, b[n - i, i, c + 1]] + b[n, i - 1, c]];
a[n_] := b[n, n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

first(n)=my(x='x+O('x^(n+1)),pr=1); concat(0,Vec(sum(j=1,n,pr*=1-x^j; j^2*x^j/pr))) \\ Charles R Greathouse IV, Dec 02 2017

G.f.: Sum_{j>=1} j^2*x^j / Product_{i=1..j} (1-x^i). - Alois P. Heinz, Dec 02 2017

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 3, 4, 1, 4, 2, 4, 3, 4, 1, 2, 4, 1, 3, 4, 2, 3, 4, 1, 2, 3, 4, 5, 1, 5, 2, 5, 3, 5, 4, 5, 1, 2, 5, 1, 3, 5, 1, 4, 5, 2, 3, 5, 2, 4, 5, 3, 4, 5, 1, 2, 3, 5, 1, 2, 4, 5, 1, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

A236515 Colexicographic index of prime signature of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 8, 2, 5, 2, 6, 4, 4, 2, 9, 3, 4, 5, 6, 2, 7, 2, 13, 4, 4, 4, 10, 2, 4, 4, 9, 2, 7, 2, 6, 6, 4, 2, 14, 3, 5, 4, 6, 2, 8, 4, 9, 4, 4, 2, 11, 2, 4, 6, 20, 4, 7, 2, 6, 4, 7, 2, 15, 2, 4, 5, 6, 4, 7, 2, 14, 8, 4, 2, 11, 4, 4, 4, 9, 2, 10, 4, 6, 4, 4, 4, 21, 2, 5, 6, 10

Offset: 1

Sung-Hyuk Cha, Jan 27 2014

nonn

Index of prime signature of n when prime signatures are listed in colexicographic order (see A036037).

			a(7) = a({1}) = 2, a(180) = a({2,2,1}) = 17, a(216) = a({3,3}) = 23
=====================================================================
a(1) = 1,
For all n in A000040, prime_signature(n) = {1}, a(n) = 2,
For all n in A001248, prime_signature(n) = {2}, a(n) = 3,
For all n in A006881, prime_signature(n) = {1,1}, a(n) = 4,
For all n in A030078, prime_signature(n) = {3}, a(n) = 5,
For all n in A054753, prime_signature(n) = {2,1}, a(n) = 6,
For all n in A007304, prime_signature(n) = {1,1,1}, a(n) = 7, etc.

OEIS Wiki, Orderings of prime signature

Cf. A036037, A124010, A237433.

A344092 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 4, 1, 3, 2, 6, 5, 1, 4, 2, 3, 2, 1, 7, 6, 1, 5, 2, 4, 3, 4, 2, 1, 8, 7, 1, 6, 2, 5, 3, 5, 2, 1, 4, 3, 1, 9, 8, 1, 7, 2, 6, 3, 5, 4, 6, 2, 1, 5, 3, 1, 4, 3, 2, 10, 9, 1, 8, 2, 7, 3, 6, 4, 7, 2, 1, 6, 3, 1, 5, 4, 1, 5, 3, 2, 4, 3, 2, 1

Offset: 0

Gus Wiseman, May 14 2021

First differs from A118457 at a(53) = 4, A118457(53) = 2.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
   0: ()
   1: (1)
   2: (2)
   3: (3)(21)
   4: (4)(31)
   5: (5)(41)(32)
   6: (6)(51)(42)(321)
   7: (7)(61)(52)(43)(421)
   8: (8)(71)(62)(53)(521)(431)
   9: (9)(81)(72)(63)(54)(621)(531)(432)

Wikiversity, Lexicographic and colexicographic order

Same as A026793 with rows reversed.
Ignoring length gives A118457.
The non-strict version is A334439 (reversed: A036036/A334302).
The version for lex instead of revlex is A344090.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A193073 reads off lexicographically ordered partitions.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A124734, A185974, A211992, A296774, A334301, A334433, A334435, A334438, A334441.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A328891 Irregular table T(n,k) = #{m > 0: m occurs m times in the k-th partition of n, using A&S order (A036036)}, 1 <= k <= A000041(n), n >= 0.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 0

M. F. Hasler, Oct 29 2019

In the n-th row, the partitions of n are considered in the "Abramowitz and Stegun" or graded (reflected or not) colexicographic ordering, as in A036036 or A036037. For each partition this counts the numbers m > 0 such that there are exactly m parts equal to m in the partition.

Row lengths are A000041(n) = number of partitions of n, the partition numbers.

			The table reads:
  n \ T(n,k), ...
  0 : 0;   (The only partition of 0 is [], having no number at all in it.)
  1 : 1;   (The only partition of 1 is [1], in which the number m=1 occurs 1 time.)
  2 : 0,0;   (Neither [2] nor [1,1] have some m occurring m times.)
  3 : 0,1,0;   ([3] and [1,1,1] have no m, but [1,2] has m=1 occurring m times.)
  4 : 0,1,1,0,0;   (Here [1,3] and [2,2] have m=1 resp. m=2 occurring m times.)
  5 : 0,1,0,0,2,0,0;   ([1,4] has m=1, [1,2,2] has m=1 and m=2 occurring m times.)
  6 : 0,1,0,0,0,1,0,0,1,0,0;
  7 : 0,1,0,0,0,1,1,1,0,0,1,0,1,0,0;
  (...)
Column 1 = (0,1,0,...) = A063524, characteristic function of {1}: The corresponding partition is [n], except for [] when n=0.
Column 2 = (0,1,1,1,...) = signum(n-2) = A057427(n-2), n >= 2: The corresponding partition is [1, n-1].
Column 3 = A063524(n-3) = A185014(n), characteristic function of {4}: The corresponding partition is [2, n-2] for n >= 4, and [1,1,1] for n = 3.
Column 4 = (0,...) = A000004(n-4), the zero function: The corresponding partition is [3, n-3] for n >= 6, and [1,1,2] for n = 4 and [1,1,3] for n = 5.
Row sums = A276428(n) = sum over all partitions of n of the number of distinct parts m of multiplicity m.

OEIS Wiki, Orderings of partitions

Cf. A036036 (list of partitions in Abramowitz & Stegun or graded reflected colexicographic order).
Cf. A000041 (partition numbers = row lengths).
Cf. A063524 (col.1: chi_{1}), A057427 (col.2: signum), A185014 (col.3: chi_{4}), A000004 (col.4: zero).
Cf. A276427 (frequency of 0, ..., max.value in each row), A276428 (row sums), A276429, A276434, A277101.
Cf. A328806 (row length of A276427(n) = 1 + largest value in row n).

apply( A328891_row(n, r=[])={forpart(p=n, my(s, c=1); for(i=1, #p, p[i]==if(i<#p, p[i+1]) && c++ && next; c==p[i] && s++; c=1); r=concat(r,s));r}, [0..12])

cp's OEIS Frontend

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

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A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

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A181087 Partitions of n in the order of increasing smallest numbers of prime signatures.

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Sage

A344091 Flattened tetrangle of all finite multisets of positive integers sorted first by sum, then by length, then colexicographically.

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A182937 Triangle in which n-th row lists all integer partitions of n, in order of traversing the periphery of the Fenner-Loizou tree in the clockwise sense.

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A296010 Sum of the squares of the number of parts in all partitions of n.

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Maple

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PARI

Formula

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

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A236515 Colexicographic index of prime signature of n.

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