A384318 -id:A384318 - cp's OEIS frontend

A179009 Number of maximally refined partitions of n into distinct parts.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 3, 5, 1, 3, 2, 3, 5, 7, 2, 5, 3, 4, 6, 7, 11, 3, 8, 5, 6, 6, 8, 11, 15, 7, 13, 9, 9, 9, 10, 12, 16, 22, 11, 20, 15, 17, 14, 15, 16, 18, 24, 30, 18, 30, 26, 28, 22, 27, 21, 25, 27, 33, 42, 36, 45, 43, 46, 38, 44, 33, 43, 36, 44, 47, 60, 46, 66, 64, 70, 63, 72, 61, 69, 60, 63, 58, 69, 80

Offset: 0

David S. Newman, Jan 03 2011

nonn

Let a_1,a_2,...,a_k be a partition of n into distinct parts. We say that this partition can be refined if one of the summands, say a_i can be replaced with two numbers whose sum is a_i  and the resulting partition is a partition into distinct parts.  For example, the partition 5+2 can be refined because 5 can be replaced by 4+1 to give 4+2+1.  If a partition into distinct parts cannot be refined we say that it is maximally refined.
The value of a(0) is taken to be 1 as is often done when considering partitions (also, the empty partition cannot be refined).
This sequence was suggested by Moshe Shmuel Newman.
From Gus Wiseman, Jun 07 2025: (Start)
Given any strict partition, the following are equivalent:
1) The parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.
(End)

			a(11)=2 because there are two partitions of 11 which are maximally refined, namely 6+4+1 and 5+3+2+1.
From _Joerg Arndt_, Apr 23 2023: (Start)
The 15 maximally refined partitions of 35 are:
   1:    [ 1 2 3 4 5 6 14 ]
   2:    [ 1 2 3 4 5 7 13 ]
   3:    [ 1 2 3 4 5 8 12 ]
   4:    [ 1 2 3 4 5 9 11 ]
   5:    [ 1 2 3 4 6 7 12 ]
   6:    [ 1 2 3 4 6 8 11 ]
   7:    [ 1 2 3 4 6 9 10 ]
   8:    [ 1 2 3 4 7 8 10 ]
   9:    [ 1 2 3 5 6 7 11 ]
  10:    [ 1 2 3 5 6 8 10 ]
  11:    [ 1 2 3 5 7 8 9 ]
  12:    [ 1 2 4 5 6 7 10 ]
  13:    [ 1 2 4 5 6 8 9 ]
  14:    [ 1 3 4 5 6 7 9 ]
  15:    [ 2 3 4 5 6 7 8 ]
(End)

Massimo Lauria, Table of n, a(n) for n = 0..1500 (first 1000 terms by Fausto A. C. Cariboni)
Riccardo Aragona, Lorenzo Campioni, Roberto Civino, and Massimo Lauria, On the maximal part in unrefinable partitions of triangular numbers, arXiv:2111.11084 [math.CO], 2021.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Riccardo Aragona, Roberto Civino, Norberto Gavioli and Carlo Maria Scoppola, Unrefinable partitions into distinct parts in a normalizer chain, arXiv:2107.04666 [math.CO], 2021.
Riccardo Aragona, Lorenzo Campioni, Roberto Civino and Massimo Lauria, Verification and generation of unrefinable partitions, arXiv:2112.15096 [math.CO], 2021.
Joerg Arndt, C++ program to compute such partitions.

For subsets instead of partitions we have A326080, complement A384350.
These partitions are ranked by A383707, apparently positions of 1 in A383706.
The strict complement is A384318 (strict partitions that can be refined).
This is the strict version of A384392, ranks A384320, complement apparently A384321.
Cf. A048767, A098859, A179822, A239455, A317142, A326083, A351293, A357982, A381454, A382525, A383708, A383710, A384317.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]=={}&]],{n,0,15}] (* Gus Wiseman, Jun 09 2025 *)

More terms from Joerg Arndt, Jan 04 2011

A384321 Numbers whose distinct prime indices are not maximally refined.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The prime indices of 25 are {3,3}, which has refinements: ((3),(1,2)) and ((1,2),(3)), so 25 is in the sequence.
The prime indices of 102 are {1,2,7}, which has refinement ((1),(2),(3,4)), so 102 is in the sequence.
The terms together with their prime indices begin:
     5: {3}      39: {2,6}      73: {21}
     7: {4}      41: {13}       74: {1,12}
    11: {5}      43: {14}       77: {4,5}
    13: {6}      46: {1,9}      79: {22}
    17: {7}      47: {15}       82: {1,13}
    19: {8}      49: {4,4}      83: {23}
    21: {2,4}    51: {2,7}      85: {3,7}
    22: {1,5}    53: {16}       86: {1,14}
    23: {9}      55: {3,5}      87: {2,10}
    25: {3,3}    57: {2,8}      89: {24}
    26: {1,6}    58: {1,10}     91: {4,6}
    29: {10}     59: {17}       93: {2,11}
    31: {11}     61: {18}       94: {1,15}
    33: {2,5}    62: {1,11}     95: {3,8}
    34: {1,7}    65: {3,6}      97: {25}
    35: {3,4}    67: {19}      101: {26}
    37: {12}     69: {2,9}     102: {1,2,7}
    38: {1,8}    71: {20}      103: {27}

These appear to be positions of terms > 1 in A383706, non-disjoint A357982, non-strict A299200.
The strict complement is A383707, counted by A179009.
Partitions of this type appear to be counted by A384317.
The complement is A384320.
The strict (squarefree) case appears to be A384322, counted by A384318.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A098859, A122111, A130091, A317142, A326080, A381454, A382525, A384005, A384323, A384390.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[30],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]!={}]&]

A384317 Number of integer partitions of n with more than one possible way to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 5, 5, 12, 12, 16, 19, 22, 35, 38, 48, 58, 68, 79, 110, 121, 149, 175, 207, 242, 281, 352, 397, 473

Offset: 0

Gus Wiseman, May 28 2025

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			There are two possibilities for (4,3), namely ((4),(3)) and ((4),(2,1)), so (4,3) is counted under a(7).
The a(3) = 1 through a(11) = 12 partitions:
  (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
                 (3,3)  (4,3)  (4,4)  (5,4)  (5,5)    (6,5)
                 (4,2)  (5,2)  (5,3)  (6,3)  (6,4)    (7,4)
                 (5,1)  (6,1)  (6,2)  (7,2)  (7,3)    (8,3)
                               (7,1)  (8,1)  (8,2)    (9,2)
                                             (9,1)    (10,1)
                                             (4,3,3)  (5,3,3)
                                             (4,4,2)  (5,4,2)
                                             (5,3,2)  (5,5,1)
                                             (5,4,1)  (6,3,2)
                                             (6,3,1)  (7,3,1)
                                             (7,2,1)  (8,2,1)

The case of a unique choice is A179009, ranks A383707.
The case of at least one choice is A383708, ranks A382913.
The case of no choices is A383710, ranks A382912.
The strict case is A384318, ranks A384322.
These partitions are ranked by A384321, positions of terms > 1 in A383706.
The case of a unique proper choice is A384323, ranks A384347, strict A384319.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A357982 counts choices of strict partitions of prime indices, non-strict A299200.
Cf. A098859, A381454, A382525, A383533, A383711, A384320.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pof[#]]>1&]],{n,0,30}]

a(n) = A383708(n) - A179009(n).

A384322 Heinz numbers of strict integer partitions with more than one possible way to choose disjoint strict partitions of each part, i.e., strict partitions that can be properly refined.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119, 122

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

			The strict partition (7,2,1) with Heinz number 102 can be properly refined into (4,3,2,1), so 102 is in the sequence.
The terms together with their prime indices begin:
     5: {3}      46: {1,9}      85: {3,7}
     7: {4}      47: {15}       86: {1,14}
    11: {5}      51: {2,7}      87: {2,10}
    13: {6}      53: {16}       89: {24}
    17: {7}      55: {3,5}      91: {4,6}
    19: {8}      57: {2,8}      93: {2,11}
    21: {2,4}    58: {1,10}     94: {1,15}
    22: {1,5}    59: {17}       95: {3,8}
    23: {9}      61: {18}       97: {25}
    26: {1,6}    62: {1,11}    101: {26}
    29: {10}     65: {3,6}     102: {1,2,7}
    31: {11}     67: {19}      103: {27}
    33: {2,5}    69: {2,9}     106: {1,16}
    34: {1,7}    71: {20}      107: {28}
    35: {3,4}    73: {21}      109: {29}
    37: {12}     74: {1,12}    111: {2,12}
    38: {1,8}    77: {4,5}     113: {30}
    39: {2,6}    79: {22}      114: {1,2,8}
    41: {13}     82: {1,13}    115: {3,9}
    43: {14}     83: {23}      118: {1,17}

The non-strict version for no choices appears to be A382912, count A383710, odd A383711.
The non-strict version for > 0 choice appears to be A382913, count A383708, odd A383533.
These are the squarefree positions of terms > 1 in A383706, see A357982, A299200.
The case of a unique choice is A383707, counted by A179009.
Partitions of this type are counted by A384318.
This is the strict/squarefree case of A384321, counted by A384317.
The case of a unique proper choice is A384390, counted by A384319, non-strict A384323.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A098859, A122111, A130091, A317142, A381454, A382525, A384005, A384320, A384347.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Select[Range[100],UnsameQ@@prix[#]&&Length[pof[prix[#]]]>1&]

A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 64, 66, 70, 72, 75, 78, 80, 81, 84, 90, 96, 98, 100, 105, 108, 110, 112, 120, 126, 128, 132, 135, 140, 144, 150, 156, 160, 162, 168, 180, 182, 192, 196

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

The squarefree case is A383707, counted by A179009.
The complement appears to be A384321, strict case A384322, counted by A384318.
Partitions of this type are counted by A384392.
A048767 is the Look-and-Say transform, fixed points A048768.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A383706, A357982 (non-disjoint), A299200 (non-strict).
Cf. A130091, A279375, A279790, A317142, A326080, A351294, A351295, A381454, A382525, A384390.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[20],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]

A384319 Number of strict integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 3, 1, 0, 4, 4, 4, 2, 0, 6, 7, 8, 8, 3, 2, 9, 9, 14, 13, 6, 7, 3, 15, 13, 20

Offset: 0

Gus Wiseman, May 28 2025

			For y = (5,4,2) we have choices ((5),(4),(2)) and ((5),(3,1),(2)), so y is counted under a(11).
The a(3) = 1 through a(11) = 4 partitions:
  (3)  (4)  .  (4,2)  (4,3)  (6,2)  .  (5,3,2)  (5,4,2)
               (5,1)  (5,2)            (5,4,1)  (6,3,2)
                      (6,1)            (6,3,1)  (7,3,1)
                                       (7,2,1)  (8,2,1)

The case of a unique choice is A179009, ranks A383707.
Choices of this type for each prime index are counted by A383706.
The non-strict version for at least one choice is A383708, ranks A382913.
The non-strict version for no choices is A383710, ranks A382912.
The non-strict version for more than one choice is A384317, ranks A384321.
The version for at least one choice is A384322, counted by A384318.
The non-strict version is A384323, ranks A384347.
These partitions are ranked by A384390.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non Look-and-Say or non section-sum partitions, ranks A351295 or A381433.
Cf. A098859, A279375, A299200, A317142, A357982, A381454, A383533, A383711, A384320.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[pof[#]]==2&]],{n,0,30}]

A384347 Heinz numbers of integer partitions with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

5, 7, 21, 22, 25, 26, 33, 35, 39, 49, 102, 114, 130, 147, 154, 165, 170, 175, 190, 195, 231, 238, 242, 255, 275, 285

Offset: 1

Gus Wiseman, May 27 2025

Positions of 2 in A383706.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 275 are {3,3,5}, with two ways to choose disjoint strict partitions of each part: ((3),(2,1),(5)) and ((2,1),(3),(5)). Hence 275 is in the sequence.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   25: {3,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
   49: {4,4}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  147: {2,4,4}
  154: {1,4,5}
  165: {2,3,5}

The case of no choices is A382912, counted by A383710, odd case A383711.
These are positions of 2 in A383706.
The case of no proper choices is A383707, counted by A179009.
The case of some proper choice is A384321, strict A384322, count A384317, strict A384318.
These partitions are counted by A384323, strict A384319.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A357982 counts strict partitions of prime indices, non-strict A299200.
Cf. A048767, A382525, A382771, A382857, A383533, A383708, A384320.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Select[Range[100],Length[pof[prix[#]]]==2&]

A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 3, 3, 2, 0, 6, 6, 6, 6, 4, 10, 10, 14, 16, 15, 16, 17, 20, 25, 27, 28, 37, 43, 31, 42, 44

Offset: 0

Gus Wiseman, May 30 2025

			For y = (4,3,3) we have two ways: ((4),(3),(2,1)) and ((4),(2,1),(3)), so y is counted under a(10).
The a(0) = 0 through a(15) = 10 partitions:
  .  .  .  3  4  .  33  43  44  .  433  533  543  544  554  5433
                    42  52  62     442  542  552  553  644  5442
                    51  61         532  551  633  652  662  5532
                                   541  632  732  661  833  5541
                                   631  731  741  733       6432
                                   721  821  831  832       6531
                                                            7431
                                                            7521
                                                            8421
                                                            9321

For just one choice we have A179009, ranked by A383707.
Twice-partitions of this type are counted by A279790.
For at least one choice we have A383708, odd case A383533.
For no choices we have A383710, odd case A383711.
For more than one choice we have A384317, ranked by A384321.
The strict version for at least one choice is A384318, ranked by A384322.
The strict version is A384319, ranked by A384390.
These partitions are ranked by A384347 = positions of 2 in A383706.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
A357982 counts choices of strict partitions of each prime index.
Cf. A098859, A299200, A317142, A381454, A382525, A382912, A382913, A384005.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pof[#]]==2&]],{n,0,15}]

A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

11, 13, 17, 19, 23, 25, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134

Offset: 1

Gus Wiseman, Jun 02 2025

nonn

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 275 are {3,3,5}, with a total of 2 proper choices: ((3),(2,1),(5)) and ((2,1),(3),(5)), so 275 is in the sequence.
The terms together with their prime indices begin:
    11: {5}      51: {2,7}      82: {1,13}
    13: {6}      53: {16}       83: {23}
    17: {7}      55: {3,5}      85: {3,7}
    19: {8}      57: {2,8}      86: {1,14}
    23: {9}      58: {1,10}     87: {2,10}
    25: {3,3}    59: {17}       89: {24}
    29: {10}     61: {18}       91: {4,6}
    31: {11}     62: {1,11}     93: {2,11}
    34: {1,7}    65: {3,6}      94: {1,15}
    37: {12}     67: {19}       95: {3,8}
    38: {1,8}    69: {2,9}      97: {25}
    41: {13}     71: {20}      101: {26}
    43: {14}     73: {21}      103: {27}
    46: {1,9}    74: {1,12}    106: {1,16}
    47: {15}     77: {4,5}     107: {28}
    49: {4,4}    79: {22}      109: {29}

Without "proper" we get A384321 (strict A384322), counted by A384317 (strict A384318).
The case of no choices is A384349, counted by A384348.
These are positions of terms > 1 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
Partitions of this type are counted by A384395.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A382912, A383710, A382913, A383708, A383533.
Cf. A179009, A357982, A381454, A382525, A383706, A383707, A384320, A384323.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Select[Range[100],Length[pofprop[prix[#]]]>1&]

A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 33, 81, 183, 402, 856, 1801, 3721, 7646, 15567, 31575

Offset: 0

Gus Wiseman, Jun 05 2025

Conjecture: Also the number of subsets of {1..n} such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each element.

			For the set s = {1,5} we have 5 = 2+3, so s is counted under a(5).
The a(0) = 0 through a(5) = 13 subsets:
  .  .  .  {3}  {3}    {3}
                {4}    {4}
                {2,4}  {5}
                {3,4}  {1,5}
                       {2,4}
                       {2,5}
                       {3,4}
                       {3,5}
                       {4,5}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {3,4,5}

The complement is counted by A326080, allowing repeats A326083.
For strict partitions of n instead of subsets of {1..n} we have A384318, ranks A384322.
First differences are A384391.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A179009 counts maximally refined strict partitions, ranks A383707.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A383706 counts ways to choose disjoint strict partitions of prime indices, non-disjoint A357982, non-strict A299200.
Cf. A279375, A279790, A317141, A317142, A383708, A383710, A384317, A384319, A384320, A384321.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,10}]

cp's OEIS Frontend

A179009 Number of maximally refined partitions of n into distinct parts.

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A384321 Numbers whose distinct prime indices are not maximally refined.

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A384317 Number of integer partitions of n with more than one possible way to choose disjoint strict partitions of each part.

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A384322 Heinz numbers of strict integer partitions with more than one possible way to choose disjoint strict partitions of each part, i.e., strict partitions that can be properly refined.

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A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.

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A384319 Number of strict integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

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A384347 Heinz numbers of integer partitions with exactly two possible ways to choose disjoint strict partitions of each part.

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A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

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A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

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