A381870 -id:A381870 - cp's OEIS frontend

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

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}]

A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 09 2025

nonn

First differs from A381716 at a(1728) = 5, A381716(1728) = 4.

Also the number of factorizations on n into prime powers > 1 with distinct sums of prime indices (A056239).

			The a(432) = 3 multiset partitions:
  {{2,2,2},{1,1,1,1}}
  {{1},{1,1,1},{2,2,2}}
  {{1},{2},{2,2},{1,1,1}}
Note {{2},{2,2},{1,1,1,1}} is not included, as it does not have distinct block-sums.

Without distinct block-sums we have A000688, after sums A381455 (upper), A381453 (lower).
For distinct blocks instead of sums we have A050361, after sums A381715.
For strict instead of constant we have A381633 (zeros A381806), after sums A381634.
Positions of 0 are A381636.
Taking block-sums (and sorting) gives A381716.
Other multiset partitions of prime indices:
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A001055 counts multiset partitions, 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, A213385, A293511, A299202, A300385, A317142, A381870.

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]}]];
Table[Length[Select[pfacs[n],UnsameQ@@hwt/@#&]],{n,100}]

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

A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 127, 168, 208, 267, 343, 431, 536, 676, 836, 1045, 1283, 1582, 1949, 2395, 2895, 3549, 4298, 5216, 6281, 7569, 9104, 10953, 13078, 15652, 18627, 22207, 26325, 31278, 37002, 43708, 51597, 60807, 71533, 84031

Offset: 0

Gus Wiseman, Mar 15 2025

nonn

			The partition y = (3,3,3,2,2,1,1,1,1) has only one multiset partition into a set of sets, namely {{1},{3},{1,2},{1,3},{1,2,3}}, but this does not have distinct sums, so y is counted under a(17).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279785.
For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
Normal multiset partitions of this type are counted by A381718, see A116539.
These partitions are ranked by A381806, zeros of A381634 and A381633.
The complement is counted by A381992, ranked by A382075.
For distinct blocks we have A382078, complement A382077, unique A382079.
MM-numbers of these multiset partitions (strict blocks with distinct sum) are A382201.
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.
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

A382077 Number of integer partitions of n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 171, 217, 283, 361, 449, 574, 721, 900, 1126, 1397, 1731, 2143, 2632, 3223, 3961, 4825, 5874, 7131, 8646, 10452, 12604, 15155, 18216, 21826, 26108, 31169, 37156, 44202, 52492, 62233, 73676, 87089, 102756, 121074

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240306 at a(14) = 76, A240306(14) = 77.

First differs from A381992 at a(17) = 171, A381992(17) = 170.

			For y = (3,2,2,2,1,1,1), we have the multiset partition {{1},{2},{1,2},{1,2,3}}, so y is counted under a(12).
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)

Factorizations of this type are counted by A050345.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions of this type are counted by A116539.
The MM-numbers of these multiset partitions are A302494.
Twice-partitions of this type are counted by A358914.
For distinct block-sums instead of blocks we have A381992, ranked by A382075.
The complement is counted by A382078, unique A382079.
These partitions are ranked by A382200, complement A293243.
For normal multisets instead of integer partitions we have A382214, complement A292432.
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.
Cf. A292444, A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870, A381990.

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[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]>0&]],{n,0,9}]

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

A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-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 06 2025

nonn

First differs from A050326 at a(30) = 4, A050326(30) = 5.
First differs from A339742 at a(42) = 5, A339742(42) = 4.
First differs from A381441 at a(30) = 4, A381441(30) = 5.
First differs from A381633 at a(210) = 10, A381633(210) = 12.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1 with distinct sums of prime indices (A056239).
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 con 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 with distinct block-sums 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 arrow {1,1,2} -> {4}.

			The prime indices of 120 are {1,1,2,3}, with 3 ways:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
with block-sums: {1,6}, {3,4}, {1,2,4}, so a(120) = 3.
The prime indices of 210 are {1,2,3,4}, with 12 ways:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{1},{2},{3},{4}}
with block-sums: {10}, {1,9}, {2,8}, {3,7}, {4,6}, {3,7}, {4,6}, {1,2,7}, {1,3,6}, {1,4,5}, {2,3,5}, {1,2,3,4}, of which 10 are distinct, so a(210) = 10.

Without distinct block-sums we have A381078 (lower A381454), before sums A050320.
For distinct blocks instead of sums we have A381441, before sums A050326, see A358914.
Before taking sums we had A381633.
Positions of 0 are A381806.
Positions of 1 are A381870, superset of A293511.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A001055 counts multiset partitions, 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, A116540, A213242, A213385, A213427, A299202, A300385, A317142, A317143, A318360.

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[Union[Sort[hwt/@#]&/@Select[sfacs[n],UnsameQ@@hwt/@#&]]],{n,100}]

A381715 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into distinct constant blocks.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

First differs from A050361 at a(1728) = 7, A050361(1728) = 8.

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 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into distinct constant blocks:
  {{2,2,2},{1,1,1,1,1,1}}
  {{1},{2,2,2},{1,1,1,1,1}}
  {{2},{2,2},{1,1,1,1,1,1}}
  {{1,1},{2,2,2},{1,1,1,1}}
  {{1},{2},{2,2},{1,1,1,1,1}}
  {{1},{1,1},{1,1,1},{2,2,2}}
  {{2},{1,1},{2,2},{1,1,1,1}}
  {{1},{2},{1,1},{2,2},{1,1,1}}
with sums:
  {6,6}
  {1,5,6}
  {2,4,6}
  {2,4,6}
  {1,2,4,5}
  {1,2,3,6}
  {2,2,4,4}
  {1,2,2,3,4}
of which 7 are distinct, so a(1728) = 7.

Without distinct blocks (A000688) we have A381455, lower (A355731) A381453.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
Positions of terms > 1 are A046099.
Before taking sums we had A050361.
For equal instead of distinct blocks we have A362421.
For strict instead of constant blocks we have A381441, before sums A050326.
For just distinct blocks we have A381452, before sums A045778.
For distinct sums we have A381716, before sums A381635, zeros A381636.
A001055 counts multiset partitions, 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.
Cf. A000720, A001222, A002846, A005117, A050342, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&&And@@SameQ@@@#&]]],{n,100}]

A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 126, 168, 207, 266, 343, 428, 534, 675, 832, 1039, 1279, 1575, 1933, 2381, 2881, 3524, 4269, 5179, 6237, 7525, 9033, 10860, 12969, 15512, 18475, 22005, 26105, 30973, 36642, 43325, 51078, 60184, 70769, 83152

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240309 at a(11) = 23, A240309(11) = 25.

First differs from A381990 at a(17) = 126, A381990(17) = 127.

			The partition y = (2,2,1,1,1) can be partitioned into sets in the following ways:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
But none of these is itself a set, so y is counted under a(7).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
For normal multisets see A292432, A292444, A116539.
These partitions are ranked by A293243, complement A382200.
The MM-numbers of these multiset partitions (set of sets) are A302494.
Twice-partitions of this type are counted by A358914.
For distinct sums we have A381990 (ranks A381806), complement A381992 (ranks A382075).
The complement is counted by A382077, unique A382079.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions into distinct sets, complement A050345.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870.

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[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]==0&]],{n,0,9}]

a(19)-a(50) from Bert Dobbelaere, Mar 29 2025

A382075 Numbers whose prime indices can be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84

Offset: 1

Gus Wiseman, Mar 19 2025

nonn

First differs from A212167 in having 3600.
First differs from A335433 in lacking 72.
First differs from A339741 in having 1080.
First differs from A345172 in lacking 72.
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 can be written as a product of squarefree numbers with distinct sums of prime indices.

			The prime indices of 1080 are {1,1,1,2,2,2,3}, and {{1},{2},{1,2},{1,2,3}} is a partition into a set of sets with distinct sums, so 1080 is in the sequence.

Twice-partitions of this type are counted by A279785, see also A358914.
These are positions of terms > 0 in A381633, see A321469, A381078, A381634.
For constant instead of strict blocks see A381635, A381636, A381716.
Normal multiset partitions into sets with distinct sums are counted by A381718.
The complement is A381806, counted by A381990.
The case of a unique choice is A381870, counted by A382079, see A382078.
Partitions of this type are counted by A381992.
For distinct blocks instead of block-sums we have A382200, complement A293243.
MM-numbers of multiset partitions into sets with distinct sums are A382201.
Normal multisets of this type are counted by A382216, see also A382214.
A001055 counts multiset partitions of prime indices, strict A045778.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
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. A089259, A270995, A293511, A318360, A381441, A382077.
Cf. A000720, A001222, A005117, A050345, A292432, A302494.

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]]]];
Select[Range[100],Length[Select[mps[prix[#]], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]

cp's OEIS Frontend

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

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A381635 Number of ways to partition the prime indices of n into constant blocks with distinct 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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A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

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A382077 Number of integer partitions of n that can be partitioned into a set of sets.

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A381634 Number of multisets that can be obtained by taking the sum of each block of a set multipartition (multiset of sets) of the prime indices of n with distinct block-sums.

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A381715 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into distinct constant blocks.

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A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.

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