A300383 -id:A300383 - cp's OEIS frontend

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

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}]

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).

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}]

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

A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 05 2025

nonn

First differs from A050320 at a(210) = 13, A050320(210) = 15. This comes from the set multipartitions {{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 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).
Set multipartitions 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 multipartition {1,1,2} -> {4}.

			The prime indices of 60 are {1,1,2,3}, with set multipartitions:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}
with block-sums: {1,6}, {3,4}, {1,1,5}, {1,2,4}, {1,3,3}, {1,1,2,3}, which are all different multisets, so a(60) = 6.

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

Before taking sums we had A050320, strict A050326 (zeros A293243), distinct sums A381633.
For distinct blocks we have A381441.
The lower version is A381454.
For distinct block-sums we have A381634.
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 A381717.
- 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 multipartitions: A089259, A116540, A270995, A296119, A318360.
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, A002846, A005117, A025487, A066328, A213242, A213385, A213427, A299201, 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]],SquareFreeQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]

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

A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 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.
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 a(21) = 6 multisets are: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, {2,1,1,1,1}, {1,1,1,1,1,1}.
The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
  ()  (2)   (4)     (6)       (G)                 (42)
      (11)  (22)    (33)      (88)                (411)
            (1111)  (222)     (4444)              (222)
                    (111111)  (22222222)          (2211)
                              (1111111111111111)  (21111)
                                                  (111111)

Robert Price, Table of n, a(n) for n = 1..300

Positions of 1 are A000079.
The strict case is A008966.
Before sorting we had A355731.
Choosing divisors instead of constant multisets gives A355733.
The upper version is A381455, before taking sums A000688.
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, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633, zeros A381806) see A381634.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
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, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]]]],{n,nn}]

a(A002110(n)) = A381807(n).

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.

Original entry on oeis.org

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.

A381871 Numbers whose prime indices cannot be partitioned into constant blocks having a common sum.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 110

Offset: 1

Gus Wiseman, Mar 13 2025

nonn

First differs from A383100 in lacking 108.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.
Also numbers that cannot be written as a product of prime powers with equal sums of prime indices.
Partitions of this type are counted by A381993.

			The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}

Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
Constant blocks with distinct sums: A381635, A381716.
For distinct instead of equal sums we have A381636, counted by A381717.
Partitions of this type are counted by A381993, complement A383093.
These are the positions of 0 in A381995.
A001055 counts multiset partitions of prime indices, strict A045778.
A050361 counts multiset partitions into distinct constant blocks.
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. A000720, A000961, A001222, A265947, A321469, A381633, A381715, A381719, A381806.

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]]]];
Select[Range[100],Select[mps[prix[#]],SameQ@@Total/@#&&And@@SameQ@@@#&]=={}&]

A300385 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the partition with Heinz number n to the local maximum.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 6, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 11, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 19, 1, 1, 2, 2, 1, 3, 1, 14, 2, 1, 1, 10, 1, 1, 1, 5, 1, 10, 1, 2, 1, 1, 1, 33, 1, 2, 2, 7, 1, 3, 1, 5, 3

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(36) = 6 maximal chains are the rows:
(2211)<(222)<(42)<(6)
(2211)<(411)<(42)<(6)
(2211)<(411)<(51)<(6)
(2211)<(321)<(42)<(6)
(2211)<(321)<(51)<(6)
(2211)<(321)<(33)<(6)

Antti Karttunen, Table of n, a(n) for n = 1..12288
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300384.

chc[ptn_]:=If[Length[ptn]===1,1,Total[chc/@Union[ReplaceList[ptn,{a___,x_,b___,y_,c___}:>Sort[{x+y,a,b,c},Greater]]]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[chc[Reverse[primeMS[n]]],{n,100}]

A300385(n) = if(1==n,0,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 += A300385((n/(f[i,1]*f[j,1])*prime(primepi(f[i,1])+primepi(f[j,1])))))); (s))); \\ Antti Karttunen, Oct 06 2018

memoA300385 = Map();
A300385(n) = if(1==n,0,if(bigomega(n)<=2,1,if(mapisdefined(memoA300385,n),mapget(memoA300385,n),my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A300385(prime(primepi(f[i,1])+primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA300385,n,s); (s)))); \\ (A memoized implementation). - Antti Karttunen, Oct 07 2018

a(1) = 0; for n > 1, if A001222(n) <= 2 [when n is 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. - Antti Karttunen, Oct 07 2018

More terms from Antti Karttunen, Oct 06 2018

cp's OEIS Frontend

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

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

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A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

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A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

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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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