A292432 -id:A292432 - cp's OEIS frontend

A318361 Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

1, 1, 0, 2, 0, 1, 0, 5, 1, 0, 0, 4, 0, 0, 0, 15, 0, 5, 0, 1, 0, 0, 0, 16, 0, 0, 8, 0, 0, 2, 0, 52, 0, 0, 0, 23, 0, 0, 0, 7, 0, 0, 0, 0, 5, 0, 0, 68, 0, 1, 0, 0, 0, 40, 0, 1, 0, 0, 0, 14, 0, 0, 1, 203, 0, 0, 0, 0, 0, 0, 0, 111, 0, 0, 4, 0, 0, 0, 0, 41, 80, 0, 0

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			The a(24) = 16 sets of sets with multiset union {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1,3},{1,2,4}}
  {{1,4},{1,2,3}}
  {{1},{2},{1,3,4}}
  {{1},{3},{1,2,4}}
  {{1},{4},{1,2,3}}
  {{1},{1,2},{3,4}}
  {{1},{1,3},{2,4}}
  {{1},{1,4},{2,3}}
  {{2},{1,3},{1,4}}
  {{3},{1,2},{1,4}}
  {{4},{1,2},{1,3}}
  {{1},{2},{3},{1,4}}
  {{1},{2},{4},{1,3}}
  {{1},{3},{4},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A007716, A045778, A049311, A050326, A116540, A292432, A292444, A293511.

Cf. A318283, A318284, A318286, A318287, A318360, A318361, A318370, A318371.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,90}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
count(sig)={my(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s==1, s[1]==1, count(sig(n))))} \\ Andrew Howroyd, Dec 18 2018

a(n) = A050326(A181821(n)).

a(prime(n)^k) = A188445(n, k). - Andrew Howroyd, Dec 17 2018

A293511 Numbers that can be written as a product of distinct squarefree numbers in exactly one way.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Gus Wiseman, Oct 11 2017

nonn

First differs from A212166 at a(128) = 363, A212166(128) = 360.

			360 is not in the sequence because it has two possible expressions: 2*3*6*10 or 2*6*30.

Cf. A001055, A005117, A045778, A050320, A050326, A292432, A292444, A293243.

nn=300;
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[#]]===1&]

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

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, 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&]

A292444 Number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 6, 9, 12, 17, 22, 30, 39, 50, 65, 83, 105, 131, 167, 207, 257, 317, 391, 478, 585, 708, 864, 1037, 1252, 1498

Offset: 1

Gus Wiseman, Oct 02 2017

Non-isomorphic finite multisets correspond to integer partitions. For example, the partition (3221) corresponds to the multiset {1,1,1,2,2,3,3,4}.

			Representatives of the a(7) = 6 multisets are: {1,1,1,1,1,1,1}, {1,1,1,1,1,1,2}, {1,1,1,1,1,2,2}, {1,1,1,1,1,2,3}, {1,1,1,1,2,2,2}, {1,1,1,1,2,2,3}.

Cf. A049311, A089259, A116540, A283877, A292432.

a(12)-a(30) from Bert Dobbelaere, Mar 30 2025

A339559 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of distinct parts, i.e., that are not the multiset union of any set of edges.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 3, 7, 6, 14, 14, 23, 27, 41, 47, 70, 84, 114, 141, 190, 225, 303, 370, 475, 578, 738, 890, 1131, 1368, 1698, 2058, 2549, 3048, 3759, 4505, 5495, 6574, 7966, 9483, 11450, 13606, 16307, 19351, 23116, 27297, 32470, 38293, 45346, 53342, 62939

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a non-graphical partition.

			The a(2) = 1 through a(10) = 14 partitions (empty column indicated by dot):
  11   .   22     2111   33       2221     44         3222       55
           1111          2211     4111     2222       6111       3322
                         3111     211111   3311       222111     3331
                         111111            5111       321111     4222
                                           221111     411111     4411
                                           311111     21111111   7111
                                           11111111              222211
                                                                 322111
                                                                 331111
                                                                 421111
                                                                 511111
                                                                 22111111
                                                                 31111111
                                                                 1111111111
For example, the partition y = (4,4,3,3,2,2,1,1,1,1) can be partitioned into a multiset of edges in just three ways:
  {{1,2},{1,2},{1,3},{1,4},{3,4}}
  {{1,2},{1,3},{1,3},{1,4},{2,4}}
  {{1,2},{1,3},{1,4},{1,4},{2,3}}
None of these are strict, so y is counted under a(22).

Eric Weisstein's World of Mathematics, Graphical partition.

A320894 ranks these partitions (using Heinz numbers).
A338915 allows equal pairs (x,x).
A339560 counts the complement in even-length partitions.
A339564 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A005117, A007717, A025065, A030229, A089259, A292432, A320893, A338899, A338903, A339619.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&strs[Times@@Prime/@#]=={}&]],{n,0,15}]

A027187(n) = a(n) + A339560(n).

More terms from Jinyuan Wang, Feb 14 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

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

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}]

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&]

cp's OEIS Frontend

A318361 Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

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A293511 Numbers that can be written as a product of distinct squarefree numbers in exactly one way.

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

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A292444 Number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets.

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A339559 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of distinct parts, i.e., that are not the multiset union of any set of edges.

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

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

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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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