A381454 -id:A381454 - cp's OEIS frontend

A381432 Heinz numbers of section-sum partitions. Union of A381431.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A320340, A364347, A350838 in containing 65.
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 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 terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}

Partitions of this type are counted by A239455, complement A351293.
The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
Union of A381431 (parts A381436).
The complement is A381433, conjugate A351295.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A051903, A066328, A116861, A130091, A212166, A238745, A239964, A317081, A381437.

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]}]];
Select[Range[100],MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A364348, A364537, A350845 in not containing 65.
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 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 terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  102: {1,2,7}
  105: {2,3,4}
  108: {1,1,2,2,2}

Partitions of this type are counted by A351293, complement A239455.
The conjugate is A351295, union of A048767 (parts A381440, fixed A048768, A217605).
The complement is A381432, union of A381431 (conjugate A351294, parts A381436).
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A181819, A238745, A239964, A317081, A381437.

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]}]];
Select[Range[100],!MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

12, 60, 63, 84, 120, 126, 132, 156, 204, 228, 252, 276, 300, 315, 325, 348, 372, 420, 444, 492, 504, 516, 560, 564, 588, 630, 636, 650, 660, 693, 708, 720, 732, 780, 804, 819, 840, 852, 876, 924, 931, 948, 975, 996, 1008, 1020, 1068, 1071, 1092, 1140, 1164

Offset: 1

Gus Wiseman, Mar 10 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.
Also numbers that cannot be written as a product of prime powers > 1 with distinct sums of prime indices (A056239).
Contains no squarefree numbers.
Conjecture: These are the zeros of A382876.

			The prime indices of 300 are {1,1,2,3,3}, with partitions into constant blocks:
  {{2},{1,1},{3,3}}
  {{1},{1},{2},{3,3}}
  {{2},{3},{3},{1,1}}
  {{1},{1},{2},{3},{3}}
but none of these has distinct block-sums, so 300 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  156: {1,1,2,6}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  300: {1,1,2,3,3}

More on multiset partitions into constant blocks: A006171, A279784, A295935.
These are the positions of 0 in A381635, after taking block-sums A381716.
Partitions of this type are counted by A381717.
For strict instead of constant blocks we have A381806, zeros of A381633.
For equal instead of distinct block-sums we have A381871.
A000688 counts multiset partitions into constant, see A381455 (upper), A381453 (lower).
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A050361 counts multiset partitions into distinct constant blocks, after sums A381715.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A005117, A050320, A059404, A213242, A293243, A299202, A300385, A381078, A381454, A381634.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Select[pfacs[#],UnsameQ@@hwt/@#&]=={}&]

A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 3, 0, 4, 1, 1, 0, 5, 0, 6, 0, 2, 2, 8, 0, 2, 2, 0, 0, 10, 1, 12, 0, 2, 3, 2, 0, 15, 3, 2, 0, 18, 1, 22, 0, 0, 5, 27, 0, 2, 0, 3, 0, 32, 0, 3, 0, 4, 5, 38, 0, 46, 7, 0, 0, 4, 1, 54, 0, 5, 1, 64, 0, 76, 8, 0, 0, 3, 1, 89, 0, 0, 10

Offset: 1

Gus Wiseman, May 15 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 prime indices of 25 are (3,3), for which we have choices ((3),(2,1)) and ((2,1),(3)), so a(25) = 2.
The prime indices of 91 are (4,6), for which we have choices ((4),(6)), ((4),(5,1)), ((4),(3,2,1)), ((3,1),(6)), ((3,1),(4,2)), so a(91) = 5.
The prime indices of 273 are (2,4,6), for which we have choices ((2),(4),(6)), ((2),(4),(5,1)), ((2),(3,1),(6)), so a(273) = 3.

Adding up over all integer partitions gives A279790, strict A279375.
Without disjointness we have A357982, non-strict version A299200.
For multiplicities instead of indices we have A382525.
Positions of 0 appear to be A382912, counted by A383710, odd case A383711.
Positions of positive terms are A382913, counted by A383708, odd case A383533.
Positions of 1 are A383707, counted by A179009.
The conjugate version is A384005.
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, A209229, A317141, A381454, A382771, A382876.

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

A381633 Number of ways to partition the prime indices of n into sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 5, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 09 2025

nonn

First differs from A050326 at 30, 60, 70, 90, ...
First differs from A339742 at 42, 66, 78, 84, ...
First differs from A381634 at a(210) = 12, A381634(210) = 10.
Also the number of factorizations on n into squarefree numbers > 1 with distinct sums of prime indices.
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, sum A056239.

			The A050320(60) = 6 ways to partition {1,1,2,3} into sets are:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}
Of these, only the following have distinct block-sums:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
So a(60) = 3.

Without distinct block-sums we have A050320, after sums A381078 (lower A381454).
For distinct blocks instead of sums we have A050326, after sums A381441, see A358914.
Taking block-sums (and sorting) gives A381634.
For constant instead of strict blocks we have A381635, see A381716, A381636.
Positions of 0 are A381806, superset of A293243.
Positions of 1 are A381870, superset of A293511.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A001055 count multiset partitions of prime indices, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A213242, A299202, A300385, A317142, A317143, A321469.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[sfacs[n],UnsameQ@@hwt/@#&]],{n,100}]

A381431 Heinz number of the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 10, 13, 11, 11, 16, 17, 15, 19, 14, 13, 13, 23, 20, 25, 17, 27, 22, 29, 13, 31, 32, 17, 19, 17, 25, 37, 23, 19, 28, 41, 17, 43, 26, 33, 29, 47, 40, 49, 35, 23, 34, 53, 45, 19, 44, 29, 31, 59, 26, 61, 37, 39, 64, 23, 19, 67, 38

Offset: 1

Gus Wiseman, Feb 26 2025

nonn

The image first differs from A320340, A364347, A350838 in containing a(150) = 65.
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 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.

			Prime indices of 180 are (3,2,2,1,1), with section-sum partition (6,3), so a(180) = 65.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
   7: {4}
  11: {5}
  10: {1,3}
  13: {6}
  11: {5}
  11: {5}
  16: {1,1,1,1}

The conjugate is A048767, union A351294, complement A351295, fix A048768 (count A217605).
Taking length instead of sum in the definition gives A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
The union is A381432, complement A381433.
Values appearing only once are A381434, more than once A381435.
These are the Heinz numbers of rows of A381436, conjugate A381440.
Greatest prime index of each term is A381437, counted by A381438.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A005117, A047966, A051903, A066328, A091602, A116861, A130091, A212166, 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[Times@@Prime/@egs[prix[n]],{n,100}]

A122111(a(n)) = A048767(n).

A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 5, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 5, 1, 1, 2, 5, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A050326 at a(210) = 13, A050326(210) = 15. This comes from the set systems {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a strict factorization of n into squarefree numbers > 1.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Sets of sets are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set of sets {1,1,2} -> {4}.

			The prime indices of 60 are {1,1,2,3}, with partitions into sets of sets:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
with block-sums: {1,6}, {3,4}, {1,2,4}, {1,3,3}, which are all different, so a(60) = 4.

Before taking sums we had A050326, non-strict A050320.
Positions of 0 are A293243.
Positions of 1 are A293511.
This is the strict version of A381078 (lower A381454).
For distinct block-sums (instead of blocks) we have A381634, before sums A381633.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on set systems: A050342, A116539, A279785, A296120, A318361.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort[Total/@prix/@#]&/@Select[facs[n],UnsameQ@@#&&And@@SquareFreeQ/@#&]]],{n,100}]

a(A002110(n)) = A066723(n).

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

Original entry on oeis.org

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

A381806 Numbers that cannot be written as a product of squarefree numbers with distinct sums of prime indices.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Gus Wiseman, Mar 12 2025

nonn

First differs from A212164 in having 3600.
First differs from A293243 in having 18000.
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.
Also numbers whose prime indices cannot be partitioned into a multiset of sets with distinct sums.

			There are 4 factorizations of 18000 into squarefree numbers:
  (2*2*3*5*10*30)
  (2*2*5*6*10*15)
  (2*2*10*15*30)
  (2*5*6*10*30)
but none of these has all distinct sums of prime indices, so 18000 is in the sequence.

Strongly normal multisets of this type are counted by A292444.
These are the zeros in A381633, see A050320, A321469, A381078, A381634.
For distinct blocks see A050326, A293243, A293511, A358914, A381441.
For more on set multipartitions see A089259, A116540, A270995, A296119, A318360.
For more on set multipartitions with distinct sums see A279785, A381718.
For constant instead of strict blocks we have A381636, see A381635, A381716.
Partitions of this type are counted by A381990, complement A381992.
The complement is A382075.
A001055 counts multiset partitions, strict A045778.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000688, A000720, A001222, A005117, A299202, A300385, A381454, A381870.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfics[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfics[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]]
Select[Range[nn],Length[Select[sqfics[#],UnsameQ@@hwt/@#&]]==0&]

A383710 Number of integer partitions of n such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 10, 15, 22, 29, 42, 59, 79, 108, 140, 190, 247, 324, 417, 541

Offset: 0

Gus Wiseman, May 07 2025

Also the number of integer partitions of n whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (221)    (222)     (322)      (332)
               (211)   (311)    (411)     (331)      (422)
               (1111)  (2111)   (2211)    (511)      (611)
                       (11111)  (3111)    (2221)     (2222)
                                (21111)   (3211)     (3221)
                                (111111)  (4111)     (3311)
                                          (22111)    (4211)
                                          (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

These partitions have Heinz numbers A382912.
The number of such families for each Heinz number is A383706.
The complement is counted by A383708, ranks A382913.
Without ones we have A383711, complement A383533.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
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. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381454, A383013.

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

cp's OEIS Frontend

A381432 Heinz numbers of section-sum partitions. Union of A381431.

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A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

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A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.

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A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

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A381633 Number of ways to partition the prime indices of n into sets with distinct sums.

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A381431 Heinz number of the section-sum partition of the prime indices of n.

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A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.

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

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A381806 Numbers that cannot be written as a product of squarefree numbers with distinct sums of prime indices.

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