A300385 -id:A300385 - cp's OEIS frontend

A381455 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 a multiset of constant multisets.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A000688 at a(144) = 9, A000688(144) = 10.
First differs from A295879 at a(128) = 15, A295879(128) = 13.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into prime powers > 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).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The prime indices of 36 are {1,1,2,2}, with the following 4 partitions into a multiset of constant multisets:
  {{1,1},{2,2}}
  {{1},{1},{2,2}}
  {{2},{2},{1,1}}
  {{1},{1},{2},{2}}
with block-sums: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, which are all different, so a(36) = 4.
The prime indices of 144 are {1,1,1,1,2,2}, with the following 10 partitions into a multiset of constant multisets:
  {{2,2},{1,1,1,1}}
  {{1},{2,2},{1,1,1}}
  {{2},{2},{1,1,1,1}}
  {{1,1},{1,1},{2,2}}
  {{1},{1},{1,1},{2,2}}
  {{1},{2},{2},{1,1,1}}
  {{2},{2},{1,1},{1,1}}
  {{1},{1},{1},{1},{2,2}}
  {{1},{1},{2},{2},{1,1}}
  {{1},{1},{1},{1},{2},{2}}
with block-sums: {4,4}, {1,3,4}, {2,2,4}, {2,2,4}, {1,1,2,4}, {1,2,2,3}, {2,2,2,2}, {1,1,1,1,4}, {1,1,2,2,2}, {1,1,1,1,2,2}, of which 9 are distinct, so a(144) = 9.
The a(n) partitions for n = 4, 8, 16, 32, 36, 64, 72, 128:
  (2)   (3)    (4)     (5)      (42)    (6)       (43)     (7)
  (11)  (21)   (22)    (32)     (222)   (33)      (322)    (43)
        (111)  (31)    (41)     (411)   (42)      (421)    (52)
               (211)   (221)    (2211)  (51)      (2221)   (61)
               (1111)  (311)            (222)     (4111)   (322)
                       (2111)           (321)     (22111)  (331)
                       (11111)          (411)              (421)
                                        (2211)             (511)
                                        (3111)             (2221)
                                        (21111)            (3211)
                                        (111111)           (4111)
                                                           (22111)
                                                           (31111)
                                                           (211111)
                                                           (1111111)

Robert Price, Table of n, a(n) for n = 1..1000

Before taking sums we had A000688.
Positions of 1 are A005117.
There is a chain from the prime indices of n to a singleton iff n belongs to A300273.
The lower version is A381453.
For distinct blocks we have A381715, before sum A050361.
For distinct block-sums we have A381716, before sums A381635 (zeros A381636).
Other multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326) see A381441 (upper).
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633) see A381634, A293243.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
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, A002577, A002846, A018818, A213242, A213427, A275870, A289078, A299200, A299202, A300385.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]

a(s) = 1 for any squarefree number s.

a(p^k) = A000041(k) for any prime p.

A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.

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, Mar 12 2025

nonn

First differs from A212166 in lacking 360.
First differs from A293511 in having 600.
Also numbers with a unique factorization into squarefree numbers with distinct sums of prime indices (A056239).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			For n = 600 the unique multiset partition is {{1},{1,3},{1,2,3}}. The unique factorization is 2*10*30.

Without distinct block-sums we have A000961, ones in A050320.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
For distinct blocks instead of sums we have A293511, ones in A050326.
These are the positions of ones in A381633, see A381634, A381806, A381990.
Normal multiset partitions of this type are counted by A381718, see A279785.
For constant instead of strict blocks we have A381991, ones in A381635.
A001055 counts multiset partitions of prime indices, strict A045778.
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.
A317141 counts coarsenings of prime indices, refinements A300383.
A321469 counts factorizations with distinct sums of prime indices, ones A166684.
Cf. A000720, A001222, A005117, A066328, A293243, A299202, A300385.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[Select[sfacs[#],UnsameQ@@hwt/@#&]]==1&]

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

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 5, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A050326 at a(30) = 4, A050326(30) = 5.
First differs from A339742 at a(42) = 5, A339742(42) = 4.
First differs from A381441 at a(30) = 4, A381441(30) = 5.
First differs from A381633 at a(210) = 10, A381633(210) = 12.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1 with distinct sums of prime indices (A056239).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition con be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions with distinct block-sums are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no arrow {1,1,2} -> {4}.

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

Without distinct block-sums we have A381078 (lower A381454), before sums A050320.
For distinct blocks instead of sums we have A381441, before sums A050326, see A358914.
Before taking sums we had A381633.
Positions of 0 are A381806.
Positions of 1 are A381870, superset of A293511.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A116540, A213242, A213385, A213427, A299202, A300385, A317142, A317143, A318360.

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

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

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

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

Without distinct blocks (A000688) we have A381455, lower (A355731) A381453.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
Positions of terms > 1 are A046099.
Before taking sums we had A050361.
For equal instead of distinct blocks we have A362421.
For strict instead of constant blocks we have A381441, before sums A050326.
For just distinct blocks we have A381452, before sums A045778.
For distinct sums we have A381716, before sums A381635, zeros A381636.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A002846, A005117, A050342, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

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

A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 4, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 8, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A045778 at a(24) = 4, A045778(24) = 5.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into distinct factors > 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 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 multisets are generally not transitive. For example, we have arrows: {{1},{2},{1,2}}: {1,1,2,2} -> {1,2,3} and {{1,2},{3}}: {1,2,3} -> {3,3}, but there is no set of multisets {1,1,2,2} -> {3,3}.

			The prime indices of 24 are {1,1,1,2}, with 5 partitions into a set of multisets:
  {{1,1,1,2}}
  {{1},{1,1,2}}
  {{2},{1,1,1}}
  {{1,1},{1,2}}
  {{1},{2},{1,1}}
with block-sums: {5}, {1,4}, {2,3}, {2,3}, {1,2,2}, of which 4 are distinct, so a(24) = 4.

Before taking sums we had A045778.
If each block is a set we have A381441, before sums A050326.
For distinct block-sums instead of blocks we have A381637, before sums A321469.
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 set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For set systems with distinct sums (A381633) see A381634, zeros A293243.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on sets of multisets: A261049, A317776, A317775, A296118, A318286.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
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, A001970, A002846, A066328, A213385, A213427, A299200, A299202, A300385, A317142.

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

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

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Mar 22 2025

nonn

Also numbers with a unique factorization into prime powers with distinct sums of prime indices.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

For distinct blocks instead of block-sums we have A004709, counted by A000726.
Twice-partitions of this type are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
These are the positions of 1 in A381635.
For no choices we have A381636 (zeros of A381635), counted by A381717.
For strict instead of constant blocks we have A381870, counted by A382079.
Partitions of this type (unique into constant with distinct) are counted by A382301.
Normal multiset partitions of this type are counted by A382203.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
Cf. A006171, A279784, A295935.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]

A381637 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 blocks with distinct sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 2, 4, 2, 5, 1, 3, 2, 4, 1, 5, 1, 2, 3, 3, 2, 5, 1, 5, 2, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

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 84 are {1,1,2,4}, with 7 multiset partitions into blocks with distinct sums:
  {{1,1,2,4}}
  {{1},{1,2,4}}
  {{2},{1,1,4}}
  {{1,1},{2,4}}
  {{1,2},{1,4}}
  {{1},{2},{1,4}}
  {{1},{4},{1,2}}
with block-sums: {8}, {1,7}, {2,6}, {2,6}, {3,5}, {1,2,5}, {1,3,4}, of which 6 are distinct, so a(84) = 6.

Allowing any block-sums gives A317141  (lower A300383), before sums A001055.
Before taking sums we had A321469.
For distinct blocks instead of distinct block-sums we have A381452.
If each block is a set we have A381634 (zeros A381806), before sums A381633.
For equal instead of distinct block-sums we have A381872, before sums A321455.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A001970, A002846, A045778, A066328, A213385, A299200, A299201, A299202, A300385, A317142.

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

A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 14, 19, 28, 39, 50, 70, 91, 120, 161, 203, 260, 338, 426, 556, 695, 863, 1082, 1360, 1685

Offset: 0

Gus Wiseman, Mar 26 2025

Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.

			The partition (3,2,2,2,1) can be partitioned as {{1},{2},{3},{2,2}} or {{1},{3},{2,2,2}}, so is counted under a(10).
The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (1111)  (221)    (51)      (61)
                            (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
The complement is counted by A381717, ranks A381636, zeros of A381635.
For strict instead of constant blocks we have A381992, ranks A382075.
For a unique choice we have A382301, ranks A381991.
Normal multiset partitions of this type are counted by A382203, sets A381718.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A295935, A300385, A353864, A381633, A381716, A381990, A381993, A382079, A382876.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]

A320105 If A001222(n) <= 2, a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)primepi(q)) (n/(p*q))), where p and q range over distinct primes dividing n. (See formula section for exact details.)

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 4, 1, 3, 1, 2, 2, 1, 1, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 11, 1, 1, 2, 1, 1, 3, 1, 2, 1, 3, 1, 16, 1, 1, 2, 2, 1, 3, 1, 8, 2, 1, 1, 11, 1, 1, 1, 4, 1, 10, 1, 2, 1, 1, 1, 16, 1, 2, 2, 6, 1, 3, 1, 4, 3

Offset: 1

Antti Karttunen, Oct 08 2018

nonn

This is an auxiliary function for computing A317145 with help of A064988. Note the similarity of the formula to that of A300385, with only difference being in the value of a(1) and that here we have multiplication (*) instead of addition (+) between primepi(p) and primepi(q).
From Gus Wiseman, Oct 09 2018: (Start)
Combinatorial interpretation is: In the poset of multiset partitions ordered by refinement, number of maximal chains from the n-th multiset multisystem (A302242) to the maximal multiset partition of the same multiset, assuming n is odd. For example, the a(315) = 10 maximal chains are
  {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12960
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A064988, A300385, A302242, A317145.

A320105(n) = if(bigomega(n)<=2,1,my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A320105(prime(primepi(f[i,1])*primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); (s));

memoA320105 = Map();
A320105(n) = if(bigomega(n)<=2,1,if(mapisdefined(memoA320105,n), mapget(memoA320105,n), my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A320105(prime(primepi(f[i,1])*primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA320105,n,s); (s))); \\ Memoized version.

If A001222(n) <= 2 [when n is one, a prime or semiprime], a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)*primepi(q)) * (n/(p*q))), where p ranges over all distinct primes dividing n, and q also ranges over primes dividing n, but with condition that q > p if p is a unitary prime factor of n, otherwise q >= p. Here primepi = A000720.

a(A064988(n)) = A317145(n).

A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 11, 2, 2, 1, 33, 1, 116, 1, 5, 4, 435, 1, 2, 11, 1, 2, 1832, 2, 8167, 1, 12, 33, 10, 1, 39700, 116, 37, 1, 201785, 5, 1099449, 4, 3, 435, 6237505, 1, 19, 2, 123, 11, 37406458, 1, 27, 2, 474, 1832, 232176847, 2, 1513796040

Offset: 1

Gus Wiseman, Mar 04 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(21) = 5 maximal chains are the rows:
(111111)<(21111)<(2211)<(222)<(42)
(111111)<(21111)<(2211)<(411)<(42)
(111111)<(21111)<(2211)<(321)<(42)
(111111)<(21111)<(3111)<(411)<(42)
(111111)<(21111)<(3111)<(321)<(42)

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300385.

pcovs[ptn_]:=Select[Union[Reverse/@Sort/@Join@@@Tuples[IntegerPartitions/@ptn]],Length[#]===Length[ptn]+1&];
coc[ptn_]:=coc[ptn]=If[Max[ptn]===1,1,Total[coc/@pcovs[ptn]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[coc[Reverse[primeMS[n]]],{n,50}]

cp's OEIS Frontend

A381455 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 a multiset of constant multisets.

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A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.

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

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

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A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

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A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

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A381637 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 blocks with distinct sums.

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A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

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A320105 If A001222(n) <= 2, a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)*primepi(q)) * (n/(p*q))), where p and q range over distinct primes dividing n. (See formula section for exact details.)

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A320105 If A001222(n) <= 2, a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)primepi(q)) (n/(p*q))), where p and q range over distinct primes dividing n. (See formula section for exact details.)