A048767 -id:A048767 - cp's OEIS frontend

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

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]]=={}]&]

A381437 Last part of the section-sum partition of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 1, 6, 5, 5, 1, 7, 2, 8, 1, 6, 6, 9, 1, 3, 7, 2, 1, 10, 6, 11, 1, 7, 8, 7, 3, 12, 9, 8, 1, 13, 7, 14, 1, 2, 10, 15, 1, 4, 3, 9, 1, 16, 2, 8, 1, 10, 11, 17, 1, 18, 12, 2, 1, 9, 8, 19, 1, 11, 8, 20, 1, 21, 13, 3, 1, 9, 9, 22, 1, 2

Offset: 1

Gus Wiseman, Feb 28 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.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.

Positions of first appearances are A008578.
The length of this partition is A051903.
The conjugate version is A051904.
For first instead of last part we get A066328.
These partitions are counted by A239455, complement A351293.
Positions of 1 are A360013, complement A381439.
This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).
This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).
Counting partitions by this statistic gives A381438.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[If[n==1,0,Last[egs[prix[n]]]],{n,100}]

a(n) = A055396(A381431(n)).

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&]

A383506 Number of non Wilf section-sum partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 4, 4, 7, 9, 12, 18, 25, 32, 42, 55, 64, 87, 101, 128, 147, 192, 218, 273, 314, 394, 450, 552, 631, 772, 886, 1066, 1221, 1458, 1677, 1980, 2269, 2672, 3029

Offset: 0

Gus Wiseman, May 18 2025

An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The a(4) = 1 through a(12) = 12 partitions (A=10, B=11):
  (31)  (32)  (51)  (43)  (53)    (54)  (64)    (65)    (75)
        (41)        (52)  (62)    (63)  (73)    (74)    (84)
                    (61)  (71)    (72)  (82)    (83)    (93)
                          (3311)  (81)  (91)    (92)    (A2)
                                        (631)   (A1)    (B1)
                                        (3322)  (632)   (732)
                                        (4411)  (641)   (831)
                                                (731)   (5511)
                                                (6311)  (6411)
                                                        (7311)
                                                        (63111)
                                                        (333111)

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A351592 (A384006).
The Look-and-Say case is A383511 (A383518).
These partitions are ranked by (A383514).
For Wilf instead of non Wilf we have A383519 (A383520).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A047966, A048767, A122111, A320348, A325324, A325325, A353837, A381431, A383530, A383709.

disjointDiffs[y_]:=Select[Tuples[IntegerPartitions /@ Differences[Prepend[Sort[y],0]]], UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n], disjointDiffs[#]!={} && !UnsameQ@@Length/@Split[#]&]],{n,0,15}]

A383709 Number of integer partitions of n with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 4, 4, 4, 5, 6, 5, 7, 8, 6, 8, 9, 9, 10, 9, 10, 12, 12, 11, 12, 14, 13, 14, 15, 14, 16, 16, 16, 18, 17, 17, 19, 20, 19, 19, 21, 21, 22, 22, 21, 24, 24, 23, 25, 25, 25, 26, 27, 27, 27, 28, 28, 30, 30, 28, 31, 32, 31, 32, 32, 33, 34, 34, 34

Offset: 0

Gus Wiseman, May 15 2025

nonn

Integer partitions with distinct multiplicities are called Wilf partitions.

			The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)    (3)  (4)    (5)      (6)      (7)      (8)
       (1,1)       (2,2)  (3,1,1)  (3,3)    (3,2,2)  (4,4)
                                   (4,1,1)  (3,3,1)  (3,3,2)
                                            (5,1,1)  (6,1,1)

For just distinct multiplicities we have A098859, ranks A130091, conjugate A383512.
For just distinct 0-appended differences we have A325324, ranks A325367.
For positive differences we have A383507, ranks A383532.
These partitions are ranked by A383712.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
A383534 gives 0-prepended differences by rank, see A325351.
Cf. A047966, A320348, A325325, A325349, A325368, A325388, A381431, A383506.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#]&&UnsameQ@@Differences[Append[#,0]]&]],{n,0,30}]

Ranked by A130091 /\ A325367

A384005 Number of ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, May 22 2025

nonn

			The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), and we have choices (6,1) and (4,2,1), so a(96) = 2.
The prime indices of 108 are {1,1,2,2,2}, conjugate (5,3), and we have choices (5,3), (5,2,1), (4,3,1), so a(108) = 3.

Adding up over all integer partitions gives A279790, strict A279375.
For multiplicities instead of indices we have conjugate of A382525.
The conjugate version is A383706.
Positive positions are A384010, conjugate A382913, counted by A383708, odd case A383533.
Positions of 0 are A384011.
Without disjointness we have A384179, conjugate A357982, non-strict version A299200.
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.
Cf. `A098859, A122111, A130091, A179009, A317141, A382771, A383707, A383710.

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

a(n) = A383706(A122111(n)).

A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 2, 5, 12, 14, 19, 35, 38, 55, 83, 107, 137, 209, 252, 359, 462, 612, 757, 1032, 1266, 1649, 2050, 2617, 3210, 4111, 4980, 6262, 7659, 9479, 11484, 14224, 17132, 20962, 25259, 30693, 36744, 44517, 53043, 63850, 75955, 90943, 107721, 128485

Offset: 0

Gus Wiseman, May 14 2025

nonn

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The a(0) = 0 through a(9) = 12 partitions:
  .  .  .  (21)  .  .  (42)    (421)   (431)    (63)
                       (321)   (3211)  (521)    (432)
                       (2211)          (3221)   (531)
                                       (4211)   (621)
                                       (32111)  (3321)
                                                (4221)
                                                (4311)
                                                (5211)
                                                (32211)
                                                (42111)
                                                (222111)
                                                (321111)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Negating both sides gives A383507, ranks A383532.
These partitions are ranked by A383531.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A033461, A047966, A111133, A320348, A325324, A325325, A325349, A325367, A325368, A325388, A383506.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&!UnsameQ@@Length/@Split[conj[#]]&]], {n,0,30}]

These partitions have Heinz numbers A130092 /\ A383513.

A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

Original entry on oeis.org

6, 21, 30, 36, 42, 60, 65, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 132, 133, 138, 140, 150, 154, 156, 165, 168, 174, 180, 186, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 286, 294, 300, 306, 308

Offset: 1

Gus Wiseman, May 15 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.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    6: {1,2}
   21: {2,4}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}

These partitions are counted by A383530.
Negating both sides gives A383532, counted by A383507.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A381431, A383506.

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]}]];
Select[Range[100],!UnsameQ@@Length/@Split[prix[#]] && !UnsameQ@@Length/@Split[conj[prix[#]]]&]

Equals A130092 /\ A383513.

A384006 Heinz numbers of Look-and-Say partitions without distinct multiplicities (non Wilf).

Original entry on oeis.org

216, 1000, 1296, 2744, 3375, 7776, 9261, 10000, 10648, 17576, 32400, 35937, 38416, 38880, 39304, 42875, 46656, 50625, 54000, 54432, 54872, 59319, 63504, 81000, 85536, 90000, 97336, 100000

Offset: 1

Gus Wiseman, May 19 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.
An integer partition is Wilf iff its multiplicities are all different, ranked by A130091, complement A130092.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

			The terms together with their prime indices begin:
     216: {1,1,1,2,2,2}
    1000: {1,1,1,3,3,3}
    1296: {1,1,1,1,2,2,2,2}
    2744: {1,1,1,4,4,4}
    3375: {2,2,2,3,3,3}
    7776: {1,1,1,1,1,2,2,2,2,2}
    9261: {2,2,2,4,4,4}
   10000: {1,1,1,1,3,3,3,3}
   10648: {1,1,1,5,5,5}
   17576: {1,1,1,6,6,6}
   32400: {1,1,1,1,2,2,2,2,3,3}
   35937: {2,2,2,5,5,5}
   38416: {1,1,1,1,4,4,4,4}
   38880: {1,1,1,1,1,2,2,2,2,2,3}
   39304: {1,1,1,7,7,7}
   42875: {3,3,3,4,4,4}
   46656: {1,1,1,1,1,1,2,2,2,2,2,2}
   50625: {2,2,2,2,3,3,3,3}
   54000: {1,1,1,1,2,2,2,3,3,3}
   54432: {1,1,1,1,1,2,2,2,2,2,4}
   54872: {1,1,1,8,8,8}
   59319: {2,2,2,6,6,6}
   63504: {1,1,1,1,2,2,2,2,4,4}
   81000: {1,1,1,2,2,2,2,3,3,3}
   85536: {1,1,1,1,1,2,2,2,2,2,5}
   90000: {1,1,1,1,2,2,3,3,3,3}
   97336: {1,1,1,9,9,9}
  100000: {1,1,1,1,1,3,3,3,3,3}

Ranking sequences are shown in parentheses below.
These partitions are counted by A351592.
For section-sum instead of Look-and-Say we have (A383514), counted by A383506.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A048767, A051903, A212166, A325368, A383520, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],disjointFamilies[prix[#]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]

cp's OEIS Frontend

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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A381437 Last part of the section-sum partition of the prime indices of n.

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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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A383506 Number of non Wilf section-sum partitions of n.

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A383709 Number of integer partitions of n with distinct multiplicities (Wilf) and distinct 0-appended differences.

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A384005 Number of ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

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A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.

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A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

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