A382216 -id:A382216 - cp's OEIS frontend

A293243 Numbers that cannot be written as a product of distinct squarefree numbers.

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, Oct 03 2017

nonn

First differs from A212164 at a(441).
Numbers n such that A050326(n) = 0. - Felix Fröhlich, Oct 04 2017
Includes A246547, and all numbers of the form p^a*q^b where p and q are primes, a >= 1 and b >= 3. - Robert Israel, Oct 10 2017
Also numbers whose prime indices cannot be partitioned into a set of sets. For example, the prime indices of 90 are {1,2,2,3}, and we have sets of sets: {{2},{1,2,3}}, {{1,2},{2,3}}, {{1},{2},{2,3}}, {{2},{3},{1,2}}, so 90 is not in the sequence. - Gus Wiseman, Apr 28 2025

			120 is not in the sequence because 120 = 2*6*10. 3600 is not in the sequence because 3600 = 2*6*10*30.

Robert Israel, Table of n, a(n) for n = 1..10000

These are the zeros of A050326.
Multiset partitions of this type (set of sets) are counted by A050342.
Twice-partitions of this type (set of sets) are counted by A279785, see also A358914.
Normal multisets of this type are counted by A292432, A292444, A381996, A382214.
The case of a unique choice is A293511, counted by A382079.
For distinct block-sums instead of blocks see A381806, A381990, A381992, A382075.
Partitions of this type are counted by A382078.
The complement is A382200, counted by A382077.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers.
A050345 counts factorizations partitioned into into distinct sets.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A001222, A005117, A089259, A116539, A246547, A270995, A381441, A382201, A382216.

N:= 1000: # to get all terms <= N
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
      S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
select(t -> A[t]=0, [$1..N]); # Robert Israel, Oct 10 2017

nn=500;
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[nn],Length[sqfacs[#]]===0&]

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

Original entry on oeis.org

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

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&]

A382200 Numbers that can be written as a product of distinct squarefree numbers.

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 21 2025

nonn

First differs from A339741 in having 1080.
First differs from A382075 in having 18000.
These are positions of positive terms in A050326, complement A293243.
Also numbers whose prime indices can be partitioned into distinct sets.
Differs from A212167, which does not include 18000 = 2^4*3^2*5^3, for example. - R. J. Mathar, Mar 23 2025

			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, so 1080 is in the sequence.
We have 18000 = 2*5*6*10*30, so 18000 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Twice-partitions of this type are counted by A279785, see also A358914.
Normal multisets not of this type are counted by A292432, strong A292444.
The complement is A293243, counted by A050342.
The case of a unique choice is A293511.
MM-numbers of multiset partitions into distinct sets are A302494.
For distinct block-sums instead of blocks we have A382075, counted by A381992.
Partitions of this type are counted by A382077, complement A382078.
Normal multisets of this type are counted by A382214, strong A381996.
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.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A001222, A005117, A050345, A089259, A116539, A270995, A381441, A382201, A382216.

N:= 1000: # to get all terms <= N
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
      S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
remove(t -> A[t]=0, [$1..N]); # Robert Israel, Apr 21 2025

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[sqfacs[#]]>0&]

A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 210, 436, 894

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A382216 at a(9) = 210, A382216(9) = 208.

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 {1,1,1,1,2,2,3,3,3} has partition {{1},{3},{1,2},{1,3},{1,2,3}}, so is counted under a(9).
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}

Factorizations of this type are counted by A050326, distinct sums A381633.
Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
The complement is counted by A292432.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
The strong version is A381996, complement A292444.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
For distinct sums we have A382216, complement A382202.
The case of a unique choice is counted by A382458, distinct sums A382459.
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, A317532, A326517, A326519, A381990, A381992, A382428, 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],Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]

A050345 Number of ways to factor n into distinct factors with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 15, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 12, 3, 12, 1, 6, 3, 12, 1, 37, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3, 1, 31, 3, 3, 3, 13, 1, 31, 3, 6, 3, 3

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

First differs from A296120 at a(36) = 15, A296120(36) = 14. - Gus Wiseman, Apr 27 2025
Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
From _Gus Wiseman_, Apr 26 2025: (Start)
This is the number of ways to partition a factorization of n (counted by A001055) into a set of sets. For example, the a(12) = 6 choices are:
  {{2},{2,3}}
  {{2},{6}}
  {{3},{4}}
  {{2,6}}
  {{3,4}}
  {{12}}
(End)

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A050346-A050350. a(A002110)=A000258.
For multisets of multisets we have A050336.
For integer partitions we have a(p^k) = A050342(k), see A001970, A089259, A261049.
For normal multiset partitions see A116539, A292432, A292444, A381996, A382214, A382216.
The case of a unique choice (positions of 1) is A166684.
Twice-partitions of this type are counted by A358914, see A270995, A281113, A294788.
For sets of multisets we have A383310 (distinct products A296118).
For multisets of sets we have we have A383311, see A296119.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A302494 gives MM-numbers of sets of sets.
A382077 counts partitions that can be partitioned into a sets of sets, ranks A382200.
A382078 counts partitions that cannot be partitioned into a sets of sets, ranks A293243.
Cf. A000009, A002865, A005117, A045782, A279785, A293511, A296120, A316439, A381441, A382201.

facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d, Rest[Divisors[n]]}]];
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[Sum[Length[Select[mps[y], UnsameQ@@#&&And@@UnsameQ@@@#&]], {y,facs[n]}],{n,30}] (* Gus Wiseman, Apr 26 2025 *)

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - R. J. Mathar, May 25 2017
a(n) = A050346(A101296(n)). - Antti Karttunen, May 25 2017

A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 47

Offset: 0

Gus Wiseman, Mar 31 2025

First differs from A382523 at a(12) = 47, A382523(12) = 45.

We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.

			Differs from A382523 in counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5} with partition {{1},{1,2},{1,3},{1,4},{1,5},{1,2,3}}
  {1,1,1,1,2,2,2,2,3,3,3,3} with partition {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Factorizations of this type are counted by A050326, distinct sums A381633.
Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
The complement is counted by A292444.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
Weak version is A382214, complement A292432, distinct sums A382216, complement A382202.
For distinct sums we have A382523, complement A382430.
Normal multiset partitions: A034691, A035310, A116540, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A255903, A317532, A326519, A381992, A382428, A382458, A382459.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n];
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[strnorm[n], Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]], {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

A382429 Number of normal multiset partitions of weight n into sets with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 13, 26, 57, 113, 283, 854, 2401, 6998, 24072, 85061, 308956, 1190518, 4770078, 19949106, 87059592

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(6) = 13 partitions:
  {1} {12}   {123}     {1234}       {12345}         {123456}
      {1}{1} {3}{12}   {12}{12}     {24}{123}       {123}{123}
             {1}{1}{1} {14}{23}     {34}{124}       {125}{134}
                       {3}{3}{12}   {3}{12}{12}     {135}{234}
                       {1}{1}{1}{1} {5}{14}{23}     {145}{235}
                                    {3}{3}{3}{12}   {12}{12}{12}
                                    {1}{1}{1}{1}{1} {14}{14}{23}
                                                    {14}{23}{23}
                                                    {16}{25}{34}
                                                    {3}{3}{12}{12}
                                                    {5}{5}{14}{23}
                                                    {3}{3}{3}{3}{12}
                                                    {1}{1}{1}{1}{1}{1}
The corresponding factorizations:
  2  6    30     210      2310       30030
     2*2  5*6    6*6      21*30      30*30
          2*2*2  14*15    35*42      6*6*6
                 5*5*6    5*6*6      66*70
                 2*2*2*2  5*5*5*6    110*105
                          11*14*15   154*165
                          2*2*2*2*2  5*5*6*6
                                     14*14*15
                                     14*15*15
                                     26*33*35
                                     5*5*5*5*6
                                     11*11*14*15
                                     2*2*2*2*2*2

Without the common sum we have A116540 (normal set multipartitions).
Twice-partitions of this type are counted by A279788.
For common sizes instead of sums we have A317583.
Without strict blocks we have A326518, non-strict blocks A326517.
For a common length instead of sum we have A331638.
For distinct instead of equal block-sums we have A381718.
Factorizations of this type are counted by A382080.
For distinct block-sums and constant blocks we have A382203.
For constant instead of strict blocks we have A382204.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A255906, A304969, A317532.
Set multipartitions: A089259, A116539, A270995, A296119, A318360.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Constant blocks with distinct sums: A381635, A381636, A381716.
Cf. A000110, A034691, A038041, A050320, A255903, A326520, A381633, A381996, 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[#],SameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(11) from Robert Price, Mar 30 2025

a(12)-a(20) from Christian Sievers, Apr 06 2025

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

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A293243 Numbers that cannot be written as a product of distinct squarefree numbers.

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

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

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A382200 Numbers that can be written as a product of distinct squarefree numbers.

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A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

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A050345 Number of ways to factor n into distinct factors with one level of parentheses.

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A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

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

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