A382429 -id:A382429 - cp's OEIS frontend

A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

1, 1, 2, 6, 23, 106, 549, 3184, 20353, 141615, 1063399, 8554800, 73281988, 665141182, 6369920854, 64133095134, 676690490875, 7462023572238, 85786458777923, 1025956348473929, 12739037494941490

Offset: 0

Gus Wiseman, Mar 26 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(3) = 6 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,2}}
                    {{2},{1,3}}
                    {{1},{2},{3}}
The a(4) = 23 factorizations:
  2*3*6  5*30    3*30    2*30    210
         10*15   6*15    6*10    2*105
         2*5*15  2*3*15  2*3*10  3*70
         3*5*10                  5*42
                                 7*30
                                 6*35
                                 10*21
                                 2*3*35
                                 2*5*21
                                 2*7*15
                                 3*5*14
                                 2*3*5*7

For distinct blocks instead of sums we have A116539, see A050326.
Without distinct sums we have A116540 (normal set multipartitions).
Twice-partitions of this type are counted by A279785.
Without strict blocks we have A326519.
Factorizations of this type are counted by A381633.
For constant instead of strict blocks we have A382203.
For distinct sizes instead of sums we have A382428, non-strict blocks A326517.
For equal instead of distinct block-sums we have A382429, non-strict blocks A326518.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A317532, A317583, A321469, A381635, A382204, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
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[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(10)-a(11) from Robert Price, Mar 31 2025

a(12)-a(20) from Christian Sievers, Apr 05 2025

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434

Offset: 0

Gus Wiseman, Mar 29 2025

We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, A116539.
The complement is counted by A382202.
Without distinct sums we have A382214, complement A292432.
The case of a unique choice is counted by A382459, without distinct sums A382458.
For Heinz numbers: A293243, A381806, A382075, A382200.
For integer partitions: A381990, A381992, A382077, A382078.
Strong version: A382523, A382430, A381996, A292444.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
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[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]

A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 5, 8, 8, 10, 8, 15, 9, 14, 15, 17, 13, 22, 14, 25, 21, 23, 19, 34, 24, 29, 28, 37, 27, 45, 29, 44, 38, 43, 43, 59, 40, 51, 48, 69, 48, 71, 52, 73, 69, 72, 61, 93, 72, 91, 77, 99, 78, 105, 95, 119, 95, 113, 96, 146, 107, 126, 123, 151, 130

Offset: 0

Gus Wiseman, Mar 26 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(6) = 7 multiset partitions:
  {1} {11}   {111}     {1111}       {11111}         {111111}
      {1}{1} {2}{11}   {11}{11}     {2}{11}{11}     {111}{111}
             {1}{1}{1} {2}{2}{11}   {2}{2}{2}{11}   {22}{1111}
                       {1}{1}{1}{1} {1}{1}{1}{1}{1} {11}{11}{11}
                                                    {2}{2}{11}{11}
                                                    {2}{2}{2}{2}{11}
                                                    {1}{1}{1}{1}{1}{1}
The a(1) = 1 through a(7) = 5 factorizations:
  2  4    8      16       32         64           128
     2*2  3*4    4*4      3*4*4      8*8          3*4*4*4
          2*2*2  3*3*4    3*3*3*4    9*16         3*3*3*4*4
                 2*2*2*2  2*2*2*2*2  4*4*4        3*3*3*3*3*4
                                     3*3*4*4      2*2*2*2*2*2*2
                                     3*3*3*3*4
                                     2*2*2*2*2*2

Christian Sievers, Table of n, a(n) for n = 0..25000

Without a common sum we have A055887.
Twice-partitions of this type are counted by A279789.
Without constant blocks we have A326518.
For distinct block-sums and strict blocks we have A381718.
Factorizations of this type are counted by A381995.
For distinct instead of equal block-sums we have A382203.
For strict instead of constant blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A255906 counts normal multiset partitions, row sums of A317532.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A304969, A356945.
Set multipartitions: A116540, A270995, A296119, A318360.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Constant blocks with distinct sums: A381635, A381636, A381716.
Cf. A007716, A034691, A034729, A116539, A255903, A317583, A326520, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
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[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

h(s,x)=my(t=0,p=1,k=1);while(s%k==0,p*=1/(1-x^(s/k))-1;t+=p;k+=1);t
lista(n)=Vec(1+sum(s=1,n,h(s,x+O(x*x^n)))) \\ Christian Sievers, Apr 05 2025

G.f.: 1 + Sum_{s>=1} Sum_{k=1..A055874(s)} Product_{v=1..k} (1/(1-x^(s/v)) - 1). - Christian Sievers, Apr 05 2025

Terms a(16) and beyond from Christian Sievers, Apr 04 2025

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Apr 22 2025

nonn

Differs from A059404, A323055, A376250 in lacking 150.
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.
Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).

			The prime indices of 150 are {1,2,3,3}, and {{3},{3},{1,2}} is a partition into sets with a common sum, so 150 is not in the sequence.

Twice-partitions of this type (sets with a common sum) are counted by A279788.
These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.
For distinct block-sums we have A381806, counted by A381990 (complement A381992).
For constant blocks we have A381871 (zeros of A381995), counted by A381993.
Partitions of this type are counted by A381994.
These are the zeros of A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
The complement counted by A383308.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A381633 counts set systems with distinct sums, see A381634, A293243.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.

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[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Select[Range[100],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]

A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 37, 76, 159, 326, 671, 1376, 2815, 5759, 11774, 24083, 49249, 100632, 205490, 419420, 855799, 1745889, 3561867, 7268240, 14836127, 30295633, 61888616

Offset: 0

Gus Wiseman, Mar 26 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 9 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1},{2,2,2}}
                                   {{2},{1,1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{1},{2},{3},{4}}
The a(5) = 19 factorizations:
  32  2*16  2*3*27   2*3*5*25  2*3*5*7*11
      4*8   2*4*9    2*3*5*9
      2*81  2*3*8    2*3*5*49
      4*27  2*3*125  2*3*7*25
      9*8   2*9*25
      3*16  2*5*27
            5*4*9

Without distinct sums we have A055887.
Twice-partitions of this type are counted by A279786.
For distinct blocks instead of sums we have A304969.
Without constant blocks we have A326519.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A381718.
For equal instead of distinct block-sums we have A382204.
For equal block-sums and strict blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A116540, A255906, A317532.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Cf. A007716, A116539, A255903, A275780, A317583, A326517, A326518, A381633, A381636, A382216, A382428.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
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[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(14)-a(26) from Christian Sievers, Apr 04 2025

A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

Original entry on oeis.org

1, 1, 1, 6, 8, 35, 292, 673, 2818, 16956, 219772, 636748, 3768505, 20309534, 183403268, 3227600747, 12272598308, 81353466578, 561187259734, 4416808925866, 50303004612136, 1238783066956740, 5566249468690291, 44970939483601100, 330144217684933896, 3131452652308459402

Offset: 0

Gus Wiseman, Mar 29 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 8 multiset partitions:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
                  {{1},{1,2}}  {{1},{1,2,3}}
                  {{1},{2,3}}  {{1},{2,3,4}}
                  {{2},{1,2}}  {{2},{1,2,3}}
                  {{2},{1,3}}  {{2},{1,3,4}}
                  {{3},{1,2}}  {{3},{1,2,3}}
                               {{3},{1,2,4}}
                               {{4},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

For distinct sums instead of sizes we have A116539, see A050326.
Without distinct lengths we have A116540 (normal set multipartitions).
Without strict blocks we have A326517, for sum instead of size A326519.
For equal instead of distinct sizes we have A331638.
Twice-partitions of this type are counted by A358830.
For distinct sums instead of sizes we have A381718.
For equal instead of distinct sizes we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A279785, A317532, A326518, A381633, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
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[Join@@(Select[mps[#],UnsameQ@@Length/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Mar 31 2025

a(10) onwards from Andrew Howroyd, Mar 31 2025

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A292432 at a(9) = 48, A292432(9) = 46.

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {1,1,1,2,2,2}
                                          {1,1,2,2,2,2}
                                          {1,2,2,2,2,2}
                                          {1,2,2,2,2,3}
                                          {1,2,3,3,3,3}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Without distinct sums we have A292432, complement A382214.
The strongly normal version without distinct sums is A292444, complement A381996.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
The complement is counted by A382216.
The strongly normal version is A382430, complement A382460.
The case of a unique choice is counted by A382459, without distinct sums A382458.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
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[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,5}]

A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 5, 16, 17, 140, 65, 1395, 2969, 22176, 1025, 1050766, 4097, 13010328, 128268897, 637598438, 65537, 64864962683, 262145, 1676258452736, 28683380484257, 24908619669860, 4194305, 30567710172480050, 8756434134071649, 62128557507554504, 21271147396968151093

Offset: 1

Andrew Howroyd, Jan 23 2020

nonn

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
From Gus Wiseman, Apr 03 2025: (Start)
Also the number of multiset partitions such that (1) the blocks together cover an initial interval of positive integers, (2) the blocks are sets of a common size, and (3) the block-sizes sum to n. For example, the a(1) = 1 through a(4) = 16 multiset partitions are:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{1,2}}
         {{1},{2}}  {{1},{1},{2}}  {{1,2},{1,3}}
                    {{1},{2},{2}}  {{1,2},{2,3}}
                    {{1},{2},{3}}  {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{1,3},{2,4}}
                                   {{1,4},{2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{2},{3}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A330942, A331639.
For constant instead of strict blocks we have A034729.
Without equal sizes we have A116540 (normal set multipartitions).
Without strict blocks we have A317583.
For distinct instead of equal sizes we have A382428, non-strict blocks A326517.
For equal sums instead of sizes we have A382429, non-strict blocks A326518.
Normal multiset partitions: A255903, A255906, A317532, A382203, A382204, A382216.
Cf. A000110, A000670, A007716, A034691, A035310, A050320, A050326, A116539, A306319, A381718.

a(n) = Sum_{d|n} A330942(n/d, d).

a(p) = 2^(p-1) + 1 for prime p.

A382304 MM-numbers of multiset partitions into sets with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 13, 16, 17, 25, 27, 29, 31, 32, 41, 43, 47, 59, 64, 67, 73, 79, 81, 83, 101, 109, 113, 121, 125, 127, 128, 137, 139, 143, 149, 157, 163, 167, 169, 179, 181, 191, 199, 211, 233, 241, 243, 256, 257, 269, 271, 277, 283, 289, 293, 313, 317

Offset: 1

Gus Wiseman, Apr 01 2025

nonn

Also products of prime numbers of squarefree index with a common sum 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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Set partitions of this type are counted by A035470.
Twice-partitions of this type are counted by A279788.
For just strict blocks we have A302478.
For just a common sum we have A326534, distinct sums A326535.
Factorizations of this type are counted by A382080.
For distinct instead of equal sums we have A382201.
For constant instead of strict blocks we have A382215.
Normal multiset partitions of this type are counted by A382429.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A058891 counts set-systems, covering A003465, connected A323818.
Cf. A000720, A005117, A050320, A050326, A293511, A302242, A302494, A381635, A381636, A381995.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@UnsameQ@@@prix/@prix[#]&]

Equals A302478 /\ A326534.

A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 10, 13, 15, 13, 31

Offset: 0

Gus Wiseman, Apr 25 2025

Any strict partition can be partitioned into a single set, so we have a lower bound a(n) >= A000009(n).

			The multiset (3,2,2,1,1) has partition {{3},{1,2},{1,2}}, so is counted under a(9).
The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)        (54)
             (111)  (31)    (41)     (42)      (52)       (53)        (63)
                    (1111)  (11111)  (51)      (61)       (62)        (72)
                                     (222)     (421)      (71)        (81)
                                     (321)     (1111111)  (431)       (333)
                                     (2211)               (521)       (432)
                                     (111111)             (2222)      (531)
                                                          (3311)      (621)
                                                          (11111111)  (3321)
                                                                      (32211)
                                                                      (222111)
                                                                      (111111111)

Twice-partitions of this type (into sets with a common sum) are counted by A279788.
Multiset partitions of this type are ranked by A326534 /\ A302478.
For distinct instead of equal sums we have A381992, see also A382077.
The complement is counted by A381994, ranks A381719.
Partitions of prime indices of this type are counted by A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
For constant instead of strict blocks we have A383093, ranks A383014.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A050320, A293511, A321455, A358914, A381633, A381717, A381993.

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[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&SameQ@@Total/@#&]]>0&]],{n,0,10}]

cp's OEIS Frontend

A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

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A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

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A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.

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A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

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A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

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A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

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A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

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A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

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