A381453 -id:A381453 - cp's OEIS frontend

A382076 Number of integer partitions of n whose run-sums are not all equal.

0, 0, 0, 1, 1, 5, 6, 13, 15, 27, 37, 54, 64, 99, 130, 172, 220, 295, 372, 488, 615, 788, 997, 1253, 1547, 1955, 2431, 3005, 3706, 4563, 5586, 6840, 8332, 10139, 12305, 14879, 17933, 21635, 26010, 31181, 37314, 44581, 53156, 63259, 75163, 89124, 105553, 124752, 147210

Offset: 0

Gus Wiseman, Apr 02 2025

nonn

Also the number of integer partitions of n that cannot be partitioned into distinct constant multisets with a common sum. Multiset partitions of this type are ranked by A005117 /\ A326534 /\ A355743, while twice-partitions are counted by A382524, strict case of A279789.

			The partition (3,2,1,1,1) has runs ((3),(2),(1,1,1)) with sums (3,2,3) so is counted under a(8).
The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)     (43)      (53)
              (41)    (51)     (52)      (62)
              (221)   (321)    (61)      (71)
              (311)   (411)    (322)     (332)
              (2111)  (2211)   (331)     (431)
                      (21111)  (421)     (521)
                               (511)     (611)
                               (2221)    (3221)
                               (3211)    (3311)
                               (4111)    (4211)
                               (22111)   (5111)
                               (31111)   (22211)
                               (211111)  (32111)
                                         (311111)
                                         (2111111)

The complement is counted by A304442, ranks A353833.
For distinct instead of equal block-sums we have A381717.
This is the strict case of A381993, see A381995, zeros A381871.
A050361 counts factorizations into distinct prime powers, see A381715.
A304405 counts partitions with weakly decreasing run-sums, ranks A357875.
A304406 counts partitions with weakly increasing run-sums, ranks A357861.
A304428 counts partitions with strictly decreasing run-sums, ranks A357862.
A304430 counts partitions with strictly increasing run-sums, ranks A357864.
A317141 counts coarsenings of prime indices, refinements A300383.
A326534 ranks multiset partitions with a common sum.
A353837 counts partitions with distinct run-sums.
A354584 lists run-sums of weakly increasing prime indices.
A355743 ranks multiset partitions into constant blocks.
Cf. A000688, A005117, A006171, A047966, A279784, A381453, A381455, A381635, A381636, A381994, A382204.

Table[Length[Select[IntegerPartitions[n],!SameQ@@Total/@Split[#]&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

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).

A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 13, 25, 33, 54, 54, 99, 124, 166, 207, 295, 352, 488, 591, 780, 987, 1253, 1488, 1951, 2419, 2993, 3665, 4563, 5508, 6840, 8270, 10127, 12289, 14869, 17781, 21635, 25992, 31167, 37184, 44581, 53008, 63259, 75076, 89080, 105531, 124752, 146842, 173516, 204141, 239921, 281461, 329929, 385852

Offset: 0

Gus Wiseman, Mar 17 2025

nonn

			The multiset partition {{2},{2},{1,1},{1,1}} has both properties (constant blocks and common sum), so (2,2,1,1,1,1) is not counted under a(8). We can also use {{2,2},{1,1,1,1}}.
The a(3) = 1 through a(8) = 13 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (32111)
                             (211111)  (311111)

Twice-partitions of this type (constant with equal) are counted by A279789.
Multiset partitions of this type are ranked by A326534 /\ A355743.
For distinct instead of equal block-sums we have A381717.
These partitions are ranked by A381871, zeros of A381995.
For strict instead of constant blocks we have A381994, see A381719, A382080.
The strict case is A382076.
Normal multiset partitions of this type are counted by A382204.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000688, A006171, A047966, A265947, A279784, A381453, A381455, A381635, A381636, A381992.

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

a(31)-a(54) from Robert Price, Mar 31 2025

A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 19 2025

nonn

Also the number of factorizations of n into prime powers > 1 with equal sums of prime indices.

			The prime indices of 144 are {1,1,1,1,2,2}, with the following 2 multiset partitions into constant blocks with a common sum:
  {{2,2},{1,1,1,1}}
  {{2},{2},{1,1},{1,1}}
so a(144) = 2.

For just constant blocks we have A000688.
Twice-partitions of this type are counted by A279789.
For just a common sum we have A321455.
For distinct instead of equal sums we have A381635.
Positions of 0 are A381871, counted by A381993.
MM-numbers of these multiset partitions are A382215.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A353864 counts rucksack partitions, ranked by A353866.
Cf. A279784, A295935, A381453 (lower), A381455 (upper).
Cf. A000720, A000961, A001222, A006171, A265947, A381633, A381719.
Cf. A323774, A382076, A382204, A382524, A383014, A383093, A383309.

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[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[mps[prix[n]], SameQ@@Total/@#&&And@@SameQ@@@#&]],{n,100}]

A323774(n) = Sum_{A056239(k)=n} a(k). Gus Wiseman, Apr 25 2025

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}]

A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Apr 22 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, sum A056239.

			The prime indices of 36 are {1,1,2,2}, and a partition into constant blocks with a common sum is: {{2},{2},{1,1}}, so 36 is in the sequence.
The prime indices of 43200 are {1,1,1,1,1,1,2,2,2,3,3}, and a partition into constant blocks with a common sum is: {{{1,1,1,1,1,1},{2,2,2},{3,3}}}, so 43200 is in the sequence.
The prime indices of 520000 are {1,1,1,1,1,1,3,3,3,3,6} and a partition into constant blocks with a common sum is: {{1,1,1,1,1,1},{3,3},{3,3},{6}}, so 520000 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}
  40: {1,1,1,3}

Twice-partitions of this type (constant blocks with a common sum) are counted by A279789.
Includes all elements of A353833.
For distinct sums we have the complement of A381636.
For strict blocks we have the complement of A381719.
For distinct sums and strict blocks we have the complement of A381806.
The complement is A381871, counted by A381993.
These are the positions of positive terms in A381995.
Partitions of this type are counted by A383093.
Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
A001055 counts factorizations (multiset partitions of prime indices), strict A045778.
A050361 counts factorizations into distinct prime powers.
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, A321469, A381635, A381715, A381716, A381717.

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]]}];
Select[Range[100], Select[Join@@@Tuples[mce/@Split[prix[#]]], SameQ@@Total/@#&]!={}&]

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 9, 5, 9, 2, 23, 2, 11, 10, 24, 2, 33, 2, 36, 12, 15, 2, 87, 7, 17, 17, 53, 2, 96, 2, 79, 16, 21, 14, 196, 2, 23, 18, 154, 2, 166, 2, 99, 54, 27, 2, 431, 9, 85, 22, 128, 2, 303, 18, 261, 24, 33, 2, 771, 2, 35, 73, 331, 20, 422, 2, 198, 28, 216, 2, 1369

Offset: 0

Gus Wiseman, Apr 22 2025

nonn

			The partition (4,4,2,2,2,2,1,1,1,1,1,1,1,1) has two partitions into constant blocks with a common sum: {{4,4},{2,2,2,2},{1,1,1,1,1,1,1,1}} and {{4},{4},{2,2},{2,2},{1,1,1,1},{1,1,1,1}}, so is counted under a(24).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (2211)               (2222)
                                     (3111)               (22211)
                                     (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)

Twice-partitions of this type (constant with common) are counted by A279789.
Multiset partitions of this type are ranked by A383309.
The complement is counted by A381993, ranks A381871.
For sets we have the complement of A381994, see A381719, A382080.
Normal multiset partitions of this type are counted by A382203, sets A381718.
For distinct instead of equal block-sums we have A382427.
These partitions are ranked by A383014, nonzeros of A381995.
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, see A381715.
A323774 counts partitions into constant blocks with a common sum
Constant blocks with distinct sums: A381635, A381636, A381717.
Permutation with equal run-sums: A383096, A383098, A383100, A383110
Cf. A006171, A047966, A279784, A295935, A323774, A326534, A353864, A355743, A381716, A382076, A382204.

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

Multiset systems of this type have MM-numbers A383309 = A326534 /\ A355743.

Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.

More terms from Jakub Buczak, May 03 2025

A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 9, 14, 16, 25, 30, 41, 52, 69, 83, 105, 129, 164, 208, 263, 315, 388, 449, 573, 694

Offset: 0

Gus Wiseman, Mar 26 2025

			The a(4) = 3 through a(8) = 14 partitions and their unique multiset partition into constant blocks with distinct sums:
  {4}     {5}       {6}         {7}        {8}
  {22}    {1}{4}    {33}        {1}{6}     {44}
  {1}{3}  {2}{3}    {1}{5}      {2}{5}     {1}{7}
          {11}{3}   {2}{4}      {3}{4}     {2}{6}
          {1}{22}   {11}{4}     {11}{5}    {3}{5}
          {2}{111}  {11}{22}    {1}{33}    {11}{6}
                    {1}{2}{3}   {3}{22}    {2}{33}
                    {1}{11}{3}  {1}{2}{4}  {11}{33}
                                {3}{1111}  {11}{222}
                                           {1}{2}{5}
                                           {1}{3}{4}
                                           {1}{3}{22}
                                           {1}{4}{111}
                                           {1}{111}{22}

For distinct blocks instead of block-sums we have A000726, ranks A004709.
Twice-partitions of this type (constant with distinct) are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
For no choices we have A381717, ranks A381636, zeros of A381635.
The Heinz numbers of these partitions are A381991, positions of 1 in A381635.
Normal multiset partitions of this type are counted by A382203.
For at least one choice we have A382427.
For strict instead of constant blocks we have A382460, ranks A381870.
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, A293511, A295935, A353864, A381633, A381716, A381990, A381992, A381993, A382079.

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],Length[Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,10}]

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}]

A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 6, 2, 10, 3, 6, 2, 24, 2, 6, 4, 17, 2, 36, 2, 18, 4, 6, 2, 86, 3, 6, 10, 18, 2, 44, 2, 50, 4, 6, 4, 159, 2, 6, 4, 62, 2, 44, 2, 18, 30, 6, 2, 486, 3, 12, 4, 18, 2, 140, 4, 62, 4, 6, 2, 932, 2, 6, 30, 157, 4, 44, 2, 18, 4, 20, 2, 1500, 2, 6

Offset: 0

Gus Wiseman, Apr 03 2025

nonn

These are strict twice-partitions of weight n and type PRR.

			The a(1) = 1 through a(8) = 10 twice-partitions:
  (1)  (2)   (3)    (4)      (5)      (6)       (7)        (8)
       (11)  (111)  (22)     (11111)  (33)      (1111111)  (44)
                    (1111)            (222)                (2222)
                    (11)(2)           (111111)             (22)(4)
                    (2)(11)           (111)(3)             (4)(22)
                                      (3)(111)             (1111)(4)
                                                           (4)(1111)
                                                           (11111111)
                                                           (1111)(22)
                                                           (22)(1111)

Gus Wiseman, Sequences enumerating triangles of integer partitions

For distinct instead of equal block-sums we have A279786.
This is the strict case of A279789.
The orderless version is A304442, see A353833, A381995, A381871.
Multiset partitions of this type are ranked by A326534 /\ A355743 /\ A005117.
Partitions with no partition of this type are counted by A382076, strict case of A381993.
Normal multiset partitions of this type are counted by the strict case of A382204.
A006171 counts multiset partitions into constant blocks of integer partitions of n.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000005, A000040, A000688, A018818, A047966, A063834, A260685, A279784, A356065, A381453, A381455.

Table[If[n==0,1,Sum[Binomial[Length[Divisors[n/d]],d]*d!,{d,Divisors[n]}]],{n,0,100}]

a(n) = Sum_{d|n} binomial(A000005(n/d),d) * d!

cp's OEIS Frontend

A382076 Number of integer partitions of n whose run-sums are not all equal.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords