A317141 -id:A317141 - 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&]

A317142 Number of refinement-ordered pairs of strict integer partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 12, 16, 24, 37, 47, 68, 90, 123, 180, 228, 307, 408, 540, 694, 970, 1207, 1598, 2048, 2669, 3357, 4382, 5599, 7109, 8990, 11428, 14330, 18144, 22652, 28343, 35746, 44269, 55094, 68384, 84780, 104477, 129360, 158682, 195323, 240177, 293704

Offset: 0

Gus Wiseman, Jul 22 2018

nonn

If x and y are strict partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y.

This sequence is dominated by A294617 (set partitions of strict partitions).

			The a(9) = 24 refinement-ordered pairs:
    (9)<=(9)
  (5,4)<=(9)   (5,4)<=(5,4)
  (6,3)<=(9)   (6,3)<=(6,3)
  (7,2)<=(9)   (7,2)<=(7,2)
  (8,1)<=(9)   (8,1)<=(8,1)
(4,3,2)<=(9) (4,3,2)<=(5,4) (4,3,2)<=(6,3) (4,3,2)<=(7,2) (4,3,2)<=(4,3,2)
(5,3,1)<=(9) (5,3,1)<=(5,4) (5,3,1)<=(6,3) (5,3,1)<=(8,1) (5,3,1)<=(5,3,1)
(6,2,1)<=(9) (6,2,1)<=(6,3) (6,2,1)<=(7,2) (6,2,1)<=(8,1) (6,2,1)<=(6,2,1)

Robert Price, Table of n, a(n) for n = 0..70

Cf. A002846, A108917, A122768, A213427, A265947, A284640, A294617, A299201, A300383, A317141.

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[Union[Select[Sort/@Map[Total,mps[ptn],{2}],UnsameQ@@#&]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 7, 1, 14, 10, 26, 1, 55, 1, 90, 68, 167, 1, 292, 1, 482, 345, 761, 1, 1291, 266, 1949, 1518, 3091, 1, 4793, 1, 7177, 5612, 10566, 2623, 16007, 1, 22912, 18992, 33619, 1, 48529, 1, 68758, 59187, 96571, 1, 137489, 11418, 189979, 167502, 264299

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

a(n) = 1 if and only if n is prime. - Chai Wah Wu, Nov 12 2018

			The a(2) = 1 through a(9) = 10 partitions:
  (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)        (333)
               (211)            (222)                (422)       (3321)
               (1111)           (321)                (431)       (32211)
                                (2211)               (2222)      (33111)
                                (3111)               (3221)      (222111)
                                (21111)              (3311)      (321111)
                                (111111)             (4211)      (2211111)
                                                     (22211)     (3111111)
                                                     (32111)     (21111111)
                                                     (41111)     (111111111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The partition (32111) can be partitioned as ((13)(112)), and the blocks both sum to 4, so (32111) is counted under a(8).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321451, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]>1&]],{n,10}]

a(n) = A000041(n) - A321451(n).

a(26)-a(52) from Alois P. Heinz, Nov 11 2018

A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 4, 14, 8, 20, 16, 55, 22, 100, 45, 108, 64, 296, 93, 489, 145, 447, 241, 1254, 284, 1692, 487, 1492, 627, 4564, 811, 6841, 1172, 4531, 1744, 12260, 1970, 21636, 3103, 12193, 3719, 44582, 4645, 63260, 6417, 29947, 8987, 124753, 9784, 162107, 14247

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

			The a(1) = 1 through a(9) = 20 partitions:
  (1)  (2)  (3)   (4)   (5)     (6)    (7)       (8)     (9)
            (21)  (31)  (32)    (42)   (43)      (53)    (54)
                        (41)    (51)   (52)      (62)    (63)
                        (221)   (411)  (61)      (71)    (72)
                        (311)          (322)     (332)   (81)
                        (2111)         (331)     (521)   (432)
                                       (421)     (611)   (441)
                                       (511)     (5111)  (522)
                                       (2221)            (531)
                                       (3211)            (621)
                                       (4111)            (711)
                                       (22111)           (3222)
                                       (31111)           (4221)
                                       (211111)          (4311)
                                                         (5211)
                                                         (6111)
                                                         (22221)
                                                         (42111)
                                                         (51111)
                                                         (411111)
A complete list of all multiset partitions of the partition (2111) into two or more blocks is: ((1)(112)), ((2)(111)), ((11)(12)), ((1)(1)(12)), ((1)(2)(11)), ((1)(1)(1)(2)). None of these has equal block-sums, so (2111) is counted toward a(5).
On the other hand, the partition (321) can be partitioned as ((12)(3)), which has two or more blocks and equal block-sums, so (321) is not counted toward a(6).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321452, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]==1&]],{n,10}]

a(n) = A000041(n) - A321452(n).

a(33)-a(50) from Alois P. Heinz, Nov 11 2018

A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 5, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 5, 1, 1, 2, 5, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A050326 at a(210) = 13, A050326(210) = 15. This comes from the set systems {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a strict factorization of n into squarefree numbers > 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Sets of sets are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set of sets {1,1,2} -> {4}.

			The prime indices of 60 are {1,1,2,3}, with partitions into sets of sets:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
with block-sums: {1,6}, {3,4}, {1,2,4}, {1,3,3}, which are all different, so a(60) = 4.

Before taking sums we had A050326, non-strict A050320.
Positions of 0 are A293243.
Positions of 1 are A293511.
This is the strict version of A381078 (lower A381454).
For distinct block-sums (instead of blocks) we have A381634, before sums A381633.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on set systems: A050342, A116539, A279785, A296120, A318361.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort[Total/@prix/@#]&/@Select[facs[n],UnsameQ@@#&&And@@SquareFreeQ/@#&]]],{n,100}]

a(A002110(n)) = A066723(n).

A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 8, 13, 14, 18, 22, 27, 36, 41, 50, 61, 73, 86

Offset: 0

Gus Wiseman, May 07 2025

Also the number of integer partitions y of n whose normal multiset (in which i appears y_i times) is a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (4,2,1)  (7,1)    (8,1)
                                                   (4,3,1)  (4,3,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)

These partitions have Heinz numbers A382913.
Without ones we have A383533, complement A383711.
The number of such families for each Heinz number is A383706.
The complement is counted by A383710, ranks A382912.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A044813, A047966, A089259, A116540, A091602, A130091, A317141, A351013, A381441, A382771, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n], pof[#]!={}&]],{n,15}]

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&]

A381716 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 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 10 2025

nonn

First differs from A381635 at a(1728) = 4, A381635(1728) = 5.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into constant multisets with distinct sums:
  {{1,1,1,1,1,1},{2,2},{2}}
  {{1,1,1,1,1},{1},{2,2,2}}
  {{1,1,1,1,1},{1},{2,2},{2}}
  {{1,1,1,1},{1,1},{2,2,2}}
  {{1,1,1},{1,1},{1},{2,2,2}}
with block-sums: {1,5,6}, {2,4,6}, {1,2,3,6}, {1,2,4,5}, so a(1728) = 4.

Without distinct sums we have A000688, after sums A381455 (upper), A381453 (lower).
More on multiset partitions into constant blocks: A006171, A279784, A295935.
For strict instead of constant we have A381633, before sums A381634.
Before taking sums we had A381635.
Positions of 0 are A381636.
For distinct blocks instead of sums we have A381715.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A002846, A005117, A050361, A213242, A265947, A293511, A299202, A300385, A362421.

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

A383710 Number of integer partitions of n such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 10, 15, 22, 29, 42, 59, 79, 108, 140, 190, 247, 324, 417, 541

Offset: 0

Gus Wiseman, May 07 2025

Also the number of integer partitions of n whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (221)    (222)     (322)      (332)
               (211)   (311)    (411)     (331)      (422)
               (1111)  (2111)   (2211)    (511)      (611)
                       (11111)  (3111)    (2221)     (2222)
                                (21111)   (3211)     (3221)
                                (111111)  (4111)     (3311)
                                          (22111)    (4211)
                                          (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

These partitions have Heinz numbers A382912.
The number of such families for each Heinz number is A383706.
The complement is counted by A383708, ranks A382913.
Without ones we have A383711, complement A383533.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381454, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n], pof[#]=={}&]], {n,0,15}]

A381717 Number of integer partitions of n that cannot be partitioned into constant multisets with distinct block-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 3, 2, 3, 6, 7, 10, 15, 15, 28, 37, 47, 64, 71, 97, 139, 173, 215, 273, 361, 439, 551, 691, 853, 1078, 1325, 1623, 2046, 2458, 2998, 3697, 4527, 5472, 6590, 7988, 9590, 11598, 13933, 16560, 19976, 23822, 28420, 33797, 40088, 47476, 56369, 66678

Offset: 0

Gus Wiseman, Mar 16 2025

nonn

Conjecture: Also the number of integer partitions of n having no permutation with all distinct run-sums, ranked by zeros of A382876. In other words, a partition has a permutation with all distinct run-sums iff it has a multiset partition into constant blocks with all distinct block-sums, where the run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums.

			For y = (3,2,2,1) we have the multiset partition {{3},{2,2},{1}}, so y is not counted under a(8).
For y = (3,2,1,1,1) there are 3 multiset partitions into constant multisets:
  {{3},{2},{1,1,1}}
  {{3},{2},{1,1},{1}}
  {{3},{2},{1},{1},{1}}
but none of these has distinct block-sums, so y is counted under a(8).
For y = (3,3,1,1,1,1,1,1) we have multiset partitions:
  {{1},{3,3},{1,1,1,1,1}}
  {{1,1},{3,3},{1,1,1,1}}
  {{1},{1,1},{3,3},{1,1,1}}
so y is not counted under a(12).
The a(4) = 1 through a(13) = 10 partitions:
  211  .  .  3211  422    4221  6211   4322     633      5422
                   4211   5211  33211  7211     8211     6331
                   32111        42211  43211    43221    9211
                                       422111   44211    54211
                                       431111   53211    63211
                                       3221111  432111   333211
                                                4221111  432211
                                                         532111
                                                         4321111
                                                         42211111

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
These partitions are ranked by A381636, zeros of A381635.
For strict instead of constant blocks we have A381990, see A381806, A381633, A382079.
For equal instead of distinct block-sums we have A381993.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383.
A050361 counts factorizations into distinct prime powers.
Cf. A002846, A047966, A130091, A213242, A213427, A265947, A317142, A353864, A381716, A381991, A381992, A382876.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Select[Join@@@Tuples[mce/@Split[#]],UnsameQ@@Total/@#&]=={}&]],{n,0,30}]

a(37)-a(53) from Robert Price, Mar 31 2025

cp's OEIS Frontend

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

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A317142 Number of refinement-ordered pairs of strict integer partitions of n.

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A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

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A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

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A381441 Number of multisets that can be obtained by partitioning the prime indices of n into a set of sets (set system) and taking their sums.

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A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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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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A381716 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 constant blocks with distinct sums.

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