A384322 -id:A384322 - cp's OEIS frontend

A384321 Numbers whose distinct prime indices are not maximally refined.

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).

A384318 Number of strict integer partitions of n that are not maximally refined.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 4, 5, 9, 10, 13, 15, 17, 26, 29, 36, 43, 49, 57, 74, 84, 101, 118, 136, 158, 181, 219, 248, 291

Offset: 0

Gus Wiseman, May 28 2025

This is the number of strict integer partitions of n containing at least one sum of distinct non-parts.

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

			For y = (5,4,2) we have 4 = 3+1 so y is counted under a(11).
On the other hand, no part of z = (6,4,1) is a subset-sum of the non-parts {2,3,5}, so z is not counted under a(11).
The a(3) = 1 through a(11) = 10 strict partitions:
  (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
                 (4,2)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                 (5,1)  (5,2)  (6,2)  (6,3)  (7,3)    (7,4)
                        (6,1)  (7,1)  (7,2)  (8,2)    (8,3)
                                      (8,1)  (9,1)    (9,2)
                                             (5,3,2)  (10,1)
                                             (5,4,1)  (5,4,2)
                                             (6,3,1)  (6,3,2)
                                             (7,2,1)  (7,3,1)
                                                      (8,2,1)

The strict complement is A179009, ranks A383707.
The non-strict version for at least one choice is A383708, for none A383710.
The non-strict version is A384317, ranks A384321, complement A384392, ranks A384320.
These partitions are ranked by A384322.
For subsets instead of partitions we have A384350, complement A326080.
Cf. A357982, A383706 (disjoint), A384319, A384323 (non-strict).
Cf. A048767, A098859, A179822, A239455, A279375, A317142, A351293, A382525, A383533, A383711, A384391.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,30}]

a(n) = A000009(n) - A179009(n).

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]]=={}]&]

A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

5, 7, 21, 22, 26, 33, 35, 39, 102, 114, 130, 154, 165, 170, 190, 195, 231, 238, 255, 285

Offset: 1

Gus Wiseman, Jun 02 2025

nonn

By "proper" we exclude the case of all singletons, which is disjoint in the strict case.

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 strict partition (7,2,1) with Heinz number 102 can only be properly refined as ((4,3),(2),(1)), so 102 is in the sequence. The other refinement ((7),(2),(1)) is not proper.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  154: {1,4,5}
  165: {2,3,5}
  170: {1,3,7}
  190: {1,3,8}
  195: {2,3,6}
  231: {2,4,5}
  238: {1,4,7}
  255: {2,3,7}
  285: {2,3,8}

The non-proper version is A383707, counted by A179009.
Partitions of this type are counted by A384319, non-strict A384323 (ranks A384347).
This is the unique case of A384321, counted by A384317.
This is the case of a unique proper choice in A384322.
The complement is A384349 \/ A384393.
These are positions of 1 in A384389.
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 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 each prime index, non-strict A299200.
Cf. A382912, counted by A383710, odd case A383711.
Cf. A382913, counted by A383708, odd case A383533.
Cf. A098859, A122111, A130091, A279375, A279790, A317142, A351201, A381454, A384005, A384320.

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&]

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}]

A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

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, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135

Offset: 1

Gus Wiseman, Jun 03 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 102 are {1,2,7}, which has proper disjoint choice ((1),(2),(3,4)), so 102 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           27: {2,2,2}        63: {2,2,4}
     2: {1}          28: {1,1,4}        64: {1,1,1,1,1,1}
     3: {2}          30: {1,2,3}        66: {1,2,5}
     4: {1,1}        32: {1,1,1,1,1}    68: {1,1,7}
     6: {1,2}        36: {1,1,2,2}      70: {1,3,4}
     8: {1,1,1}      40: {1,1,1,3}      72: {1,1,1,2,2}
     9: {2,2}        42: {1,2,4}        75: {2,3,3}
    10: {1,3}        44: {1,1,5}        76: {1,1,8}
    12: {1,1,2}      45: {2,2,3}        78: {1,2,6}
    14: {1,4}        48: {1,1,1,1,2}    80: {1,1,1,1,3}
    15: {2,3}        50: {1,3,3}        81: {2,2,2,2}
    16: {1,1,1,1}    52: {1,1,6}        84: {1,1,2,4}
    18: {1,2,2}      54: {1,2,2,2}      88: {1,1,1,5}
    20: {1,1,3}      56: {1,1,1,4}      90: {1,2,2,3}
    24: {1,1,1,2}    60: {1,1,2,3}      92: {1,1,9}

The non-proper version appears to be A382912, counted by A383710.
The non-proper complement appears to be A382913, counted by A383708.
The complement is A384321, counted by A384317.
These partitions are counted by A384348.
These are the positions of 0 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
A048767 is the Look-and-Say transform, fixed points A048768.
A056239 adds up prime indices, row sums of A112798.
A179009 counts maximally refined strict partitions, ranks A383707.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
Cf. A122111, A130091, A317142, A326080, A351294, A357982, A381454, A382525, A383706, A384320, A384322.

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[#]]]==0&]

A384389 Number of proper ways to choose disjoint strict integer partitions of each prime index of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 0, 0, 4, 0, 5, 0, 1, 1, 7, 0, 2, 1, 0, 0, 9, 0, 11, 0, 1, 2, 1, 0, 14, 2, 1, 0, 17, 0, 21, 0, 0, 4, 26, 0, 2, 0, 2, 0, 31, 0, 2, 0, 3, 4, 37, 0, 45, 6, 0, 0, 3, 0, 53, 0, 4, 0, 63, 0, 75, 7, 0, 0, 2, 0, 88, 0, 0, 9

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.

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

			The prime indices of 65 are {3,6}, and we have proper choices: ((3),(5,1)), ((3),(4,2)), ((2,1),(6)). Hence a(65) = 3.
The prime indices of 175 are {3,3,4}, and we have choices: ((3),(2,1),(4)), ((2,1),(3),(4)), both already proper. Hence a(175) = 2.

Without disjointness we have A357982 - 1, non-strict version A299200 - 1.
This is the proper case of A383706, conjugate version A384005.
Positions of positive terms are A384321.
Positions of 0 are A384349.
Positions of 1 are A384390.
Positions of terms > 1 are A384393.
The conjugate version is A384394.
Positions of first appearances are A384396.
A000041 counts integer partitions, strict A000009.
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
A351293 counts non-Look-and-Say partitions, ranks A351295.
Cf. A382912, A383710, A383711, A382913, A383708, A383533.
Cf. A179009, A279375, A279790, A381454, A382525, A384322, A384347.

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@@#&];
Table[Length[pofprop[prix[n]]],{n,100}]

a(prime(n)) = A000009(n) - 1.

cp's OEIS Frontend

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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A384318 Number of strict integer partitions of n that are not maximally refined.

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

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A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

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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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A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

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