A381078 -id:A381078 - 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[#]]],#]&]

A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A357982 at a(25) = 3, A357982(25) = 4.
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).
Set multipartitions 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 multipartition {1,1,2} -> {4}.

			The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.

For constant instead of strict partitions see A381453, A355733, A381455, A000688.
Positions of 1 are A003586.
The upper version is A381078, before sums A050320.
For distinct block-sums see A381634, A381633, A381806.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set systems (A050326, zeros A293243) see A381441 (upper).
- 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, zeros A381636) see A381716.
More on set systems: A050342, A116539, A296120, A318361.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
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.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.

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

a(A002110(n)) = A381808(n).

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/@#&]=={}&]

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).

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&]

A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 170, 217, 282, 360, 449, 571, 719, 899, 1122, 1391, 1727, 2136, 2616, 3209, 3947, 4800, 5845, 7094, 8602, 10408, 12533, 15062, 18107, 21686, 25956, 30967, 36936, 43897, 52132, 61850, 73157, 86466, 101992, 120195

Offset: 0

Gus Wiseman, Mar 16 2025

nonn

Also the number of integer partitions of n whose Heinz number belongs to A382075 (can be written as a product of squarefree numbers with distinct sums of prime indices).

			There are 6 ways to partition (3,2,2,1) into sets:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
  {{2},{2},{1,3}}
  {{2},{3},{1,2}}
  {{1},{2},{2},{3}}
Of these, 3 have distinct block sums:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
so (3,2,2,1) is counted under a(8).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633, zeros of A381634.
For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
Normal multiset partitions of this type are counted by A381718, see A116539.
The complement is counted by A381990, ranked by A381806.
These partitions are ranked by A382075.
For distinct blocks instead of sums we have A382077, complement A382078.
For a unique choice we have A382079.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
A382201 lists MM-numbers of sets with distinct sums.
Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
Cf. A293243, A293511, A358914, A381078, A381441, A381454.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,10}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 4, 1, 2, 3, 4, 2, 6, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 1, 4, 2, 6, 3, 6, 4, 8, 2, 2, 6, 4, 2, 6, 3, 4, 2, 6, 3, 4, 4, 5, 4, 4, 3, 8, 4, 2, 4, 6, 2, 8, 1, 8, 4, 2, 2, 6, 6, 6, 3, 4, 6, 6, 4, 6, 8, 4, 2, 5, 2, 2, 6, 4, 4, 8

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 5.
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).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The a(21) = 6 multisets are: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, {2,1,1,1,1}, {1,1,1,1,1,1}.
The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
  ()  (2)   (4)     (6)       (G)                 (42)
      (11)  (22)    (33)      (88)                (411)
            (1111)  (222)     (4444)              (222)
                    (111111)  (22222222)          (2211)
                              (1111111111111111)  (21111)
                                                  (111111)

Robert Price, Table of n, a(n) for n = 1..300

Positions of 1 are A000079.
The strict case is A008966.
Before sorting we had A355731.
Choosing divisors instead of constant multisets gives A355733.
The upper version is A381455, before taking sums A000688.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633, zeros A381806) see A381634.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
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, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]]]],{n,nn}]

a(A002110(n)) = A381807(n).

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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A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

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

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

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A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.

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