A384347 -id:A384347 - cp's OEIS frontend

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

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&]

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}]

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}]

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.

A384348 Number of integer partitions of n with no proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 7, 11, 17, 25, 30, 44, 61, 82, 113, 141, 193, 249, 327, 422, 548, 682, 881, 1106, 1400, 1751

Offset: 0

Gus Wiseman, May 30 2025

nonn

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

			For the partition y = (5,4,2,1) we have the following proper ways to choose strict partitions of each part:
  ((5),(3,1),(2),(1))
  ((4,1),(4,2),(1))
  ((4,1),(3,1),(2),(1))
  ((3,2),(4),(2),(1))
  ((3,2),(3,1),(2),(1))
But none of this is disjoint, so y is counted under a(12).
The a(1) = 1 through a(8) = 17 partitions:
  (1)  (2)   (21)   (22)    (32)     (222)     (322)      (332)
       (11)  (111)  (31)    (41)     (321)     (331)      (422)
                    (211)   (221)    (411)     (421)      (431)
                    (1111)  (311)    (2211)    (511)      (521)
                            (2111)   (3111)    (2221)     (611)
                            (11111)  (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (3311)
                                               (31111)    (4211)
                                               (211111)   (5111)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The strict case is A179009, ranked by A383707.
This is the proper version of A383710, odd case A383711.
This is the proper complement of A383708, odd case A383533.
The complement is counted by A384317, ranks A384321.
The strict version for at least one proper choice is A384318, ranked by A384322.
For just one proper choice we have A384319, ranked by A384390.
For two choices we have A384323, ranks A384347 = positions of 2 in A383706.
These partitions are ranked by A384349.
For more than one proper choice we have A384395, ranked by A384393.
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.
Cf. A098859, A279790, A299200, A317142, A357982, A381454, A382525, A382912, A382913, A384005.

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

A384394 Number of proper ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 03 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.

			The prime indices of 216 are {1,1,1,2,2,2}, with conjugate partition (6,3), with proper choices ((6),(2,1)), ((5,1),(3)), and ((4,2),(3)), so a(216) = 3.

Conjugate prime indices are the rows of A122111.
The non-proper version is A384005, conjugate A383706.
This is the conjugate version of A384389 (firsts 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 or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
See also A382912, counted by A383710, odd case A383711.
See also A382913, counted by A383708, odd case A383533.
Cf. A279790, A357982, A381454, A382525, A384321, A384322, A384347, A384349, A384390, A384393.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Table[Length[pofprop[conj[prix[n]]]],{n,100}]

A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 8, 8, 12, 17, 22, 29, 31, 40, 50, 65, 77, 101, 112, 135, 162, 201

Offset: 0

Gus Wiseman, May 30 2025

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

			For the partition (8,5,2) we have four choices:
  ((8),(4,1),(2))
  ((7,1),(5),(2))
  ((5,3),(4,1),(2))
  ((4,3,1),(5),(2))
Hence (8,5,2) is counted under a(15).
The a(5) = 1 through a(12) = 12 partitions:
  (5)  (6)    (7)  (8)    (9)    (10)     (11)     (12)
       (3,3)       (4,4)  (5,4)  (5,5)    (6,5)    (6,6)
                   (5,3)  (6,3)  (6,4)    (7,4)    (7,5)
                   (7,1)  (7,2)  (7,3)    (8,3)    (8,4)
                          (8,1)  (8,2)    (9,2)    (9,3)
                                 (9,1)    (10,1)   (10,2)
                                 (4,3,3)  (5,3,3)  (11,1)
                                 (4,4,2)  (5,5,1)  (5,5,2)
                                                   (6,3,3)
                                                   (6,4,2)
                                                   (6,5,1)
                                                   (9,2,1)

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 at least one proper choice we have A384317, ranked by A384321.
The strict version for at least one proper choice is A384318, ranked by A384322.
The strict version for just one proper choice is A384319, ranked by A384390.
For just one proper choice we have A384323, ranks A384347 = positions of 2 in A383706.
For no proper choices we have A384348, ranked by A384349.
These partitions are ranked by A384393.
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, non-strict A299200.
Cf. A098859, A279375, A317142, A381454, A382525, A382912, A382913, A384005.

pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pofprop[#]]>1&]],{n,0,15}]

A384396 Position of first appearance of n in A384389 (proper choices of disjoint strict partitions of each prime index).

Original entry on oeis.org

1, 5, 11, 13, 17, 19, 62, 23, 111, 29, 123, 31, 129, 217, 37, 141, 106, 41, 159, 391, 118, 43

Offset: 0

Gus Wiseman, Jun 03 2025

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.

Positions of first appearances in A384389.
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, A382913, A383708.
Cf. A179009, A279790, A357982, A381454, A382525, A383706, A384321, A384322, A384347, A384349, A384390, A384393.

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@@#&];
lv=Table[Length[pofprop[prix[n]]],{n,100}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[lv,x][[1,1]],{x,0,mnrm[lv+1]-1}]

cp's OEIS Frontend

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

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

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A384348 Number of integer partitions of n with no proper way to choose disjoint strict partitions of each part.

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A384394 Number of proper ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

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A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

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