A296119 -id:A296119 - cp's OEIS frontend

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.

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).

A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 127, 168, 208, 267, 343, 431, 536, 676, 836, 1045, 1283, 1582, 1949, 2395, 2895, 3549, 4298, 5216, 6281, 7569, 9104, 10953, 13078, 15652, 18627, 22207, 26325, 31278, 37002, 43708, 51597, 60807, 71533, 84031

Offset: 0

Gus Wiseman, Mar 15 2025

nonn

			The partition y = (3,3,3,2,2,1,1,1,1) has only one multiset partition into a set of sets, namely {{1},{3},{1,2},{1,3},{1,2,3}}, but this does not have distinct sums, so y is counted under a(17).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279785.
For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
Normal multiset partitions of this type are counted by A381718, see A116539.
These partitions are ranked by A381806, zeros of A381634 and A381633.
The complement is counted by A381992, ranked by A382075.
For distinct blocks we have A382078, complement A382077, unique A382079.
MM-numbers of these multiset partitions (strict blocks with distinct sum) are A382201.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
Cf. A293243, A293511, A358914, A381078, A381441, A381454.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,10}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A296120 Number of ways to choose a strict factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 14, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 11, 3, 12, 1, 6, 3, 12, 1, 36, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(36) = 14 twice-factorizations:
(36), (4*9), (3*12), (2*18), (2*3*6),
(4)*(9), (3)*(12), (3)*(3*4), (3)*(2*6), (2)*(18), (2)*(3*6), (2)*(2*9),
(2)*(3)*(6), (2)*(3)*(2*3).

Cf. A000009, A005117, A045778, A050345, A279785, A281113, A294788, A296118, A296119, A296121.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@Length/@sfs/@fac,{fac,sfs[n]}],{n,100}]

Dirichlet g.f.: Product_{n > 1}(1 + A045778(n)/n^s).

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&]

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

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 171, 217, 283, 361, 449, 574, 721, 900, 1126, 1397, 1731, 2143, 2632, 3223, 3961, 4825, 5874, 7131, 8646, 10452, 12604, 15155, 18216, 21826, 26108, 31169, 37156, 44202, 52492, 62233, 73676, 87089, 102756, 121074

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240306 at a(14) = 76, A240306(14) = 77.

First differs from A381992 at a(17) = 171, A381992(17) = 170.

			For y = (3,2,2,2,1,1,1), we have the multiset partition {{1},{2},{1,2},{1,2,3}}, so y is counted under a(12).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)

Factorizations of this type are counted by A050345.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions of this type are counted by A116539.
The MM-numbers of these multiset partitions are A302494.
Twice-partitions of this type are counted by A358914.
For distinct block-sums instead of blocks we have A381992, ranked by A382075.
The complement is counted by A382078, unique A382079.
These partitions are ranked by A382200, complement A293243.
For normal multisets instead of integer partitions we have A382214, complement A292432.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A292444, A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870, A381990.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]>0&]],{n,0,9}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

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

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 126, 168, 207, 266, 343, 428, 534, 675, 832, 1039, 1279, 1575, 1933, 2381, 2881, 3524, 4269, 5179, 6237, 7525, 9033, 10860, 12969, 15512, 18475, 22005, 26105, 30973, 36642, 43325, 51078, 60184, 70769, 83152

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240309 at a(11) = 23, A240309(11) = 25.

First differs from A381990 at a(17) = 126, A381990(17) = 127.

			The partition y = (2,2,1,1,1) can be partitioned into sets in the following ways:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
But none of these is itself a set, so y is counted under a(7).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
For normal multisets see A292432, A292444, A116539.
These partitions are ranked by A293243, complement A382200.
The MM-numbers of these multiset partitions (set of sets) are A302494.
Twice-partitions of this type are counted by A358914.
For distinct sums we have A381990 (ranks A381806), complement A381992 (ranks A382075).
The complement is counted by A382077, unique A382079.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions into distinct sets, complement A050345.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]==0&]],{n,0,9}]

a(19)-a(50) from Bert Dobbelaere, Mar 29 2025

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434

Offset: 0

Gus Wiseman, Mar 29 2025

We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, A116539.
The complement is counted by A382202.
Without distinct sums we have A382214, complement A292432.
The case of a unique choice is counted by A382459, without distinct sums A382458.
For Heinz numbers: A293243, A381806, A382075, A382200.
For integer partitions: A381990, A381992, A382077, A382078.
Strong version: A382523, A382430, A381996, A292444.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]

A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 4, 1, 1, 1, 2, 2, 4, 5, 3, 1, 6, 5, 5, 1, 1, 1, 1, 7, 3, 2, 8, 4, 1, 6, 6, 9, 3, 1, 1, 3, 3, 7, 2, 2, 2, 5, 1, 10, 6, 11, 1, 1, 1, 1, 1, 7, 8, 7, 3, 3, 12, 9, 8, 4, 1, 1, 13, 7, 14, 6, 1, 5, 2, 10, 15, 3, 1, 1, 1, 4, 4, 4, 3, 9, 7, 1, 16, 3, 2, 2

Offset: 1

Gus Wiseman, Feb 28 2025

Row-lengths are A051903.
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 section-sum partition of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 24 are (2,1,1,1), with sections ((2,1),(1),(1)), so row 24 is (3,1,1).
Triangle begins:
   1: (empty)
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 3
   7: 4
   8: 1 1 1
   9: 2 2
  10: 4
  11: 5
  12: 3 1
  13: 6
  14: 5
  15: 5
  16: 1 1 1 1

Row-lengths are A051903.
Row sums are A056239.
First part in each row is A066328.
Taking length instead of sum gives A238744, Heinz numbers A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
Heinz numbers are A381431 (union A381432, complement A381433, fixed A000961, A000005).
Rows appearing only once have Heinz numbers A381434, more than once A381435.
Last part in each row is A381437, counted by A381438.
The conjugate is A381440, Heinz numbers A048767 (union A351294, complement A351295).
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.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A001222, A003557, A005117, A047966, A116861, A130091, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[egs[prix[n]],{n,100}]

A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 28 2025

Row lengths are A066328.
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 Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The prime indices of 24 are (2,1,1,1), with Look-and-Say partition (3,1,1), so row 24 is (3,1,1).
The prime indices of 36 are (2,2,1,1), with Look-and-Say partition (2,2,2), so row 36 is (2,2,2).
Triangle begins:
   1: (empty)
   2: 1
   3: 1 1
   4: 2
   5: 1 1 1
   6: 1 1 1
   7: 1 1 1 1
   8: 3
   9: 2 2
  10: 1 1 1 1
  11: 1 1 1 1 1
  12: 2 1 1
  13: 1 1 1 1 1 1
  14: 1 1 1 1 1
  15: 1 1 1 1 1
  16: 4
  17: 1 1 1 1 1 1 1
  18: 2 2 1
  19: 1 1 1 1 1 1 1 1

Heinz numbers are A048767 (union A351294, complement A351295, fixed A048768, A217605).
First part in each row is A051903, conjugate A066328.
Last part in each row is A051904, conjugate A381437 (counted by A381438).
Row sums are A056239.
Row lengths are A066328.
Partitions of this type are counted by A239455, complement A351293.
The conjugate is A381436, Heinz numbers A381431 (union A381432, complement A381433).
Rows appearing only once have Heinz numbers A381540, more than once A381541.
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, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A071178, A091602, A116861, A130091, A181819, A212166, A238744, A380955.

Table[Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>ConstantArray[k,PrimePi[p]]]]//Reverse,{n,30}]

A381437 Last part of the section-sum partition of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 1, 6, 5, 5, 1, 7, 2, 8, 1, 6, 6, 9, 1, 3, 7, 2, 1, 10, 6, 11, 1, 7, 8, 7, 3, 12, 9, 8, 1, 13, 7, 14, 1, 2, 10, 15, 1, 4, 3, 9, 1, 16, 2, 8, 1, 10, 11, 17, 1, 18, 12, 2, 1, 9, 8, 19, 1, 11, 8, 20, 1, 21, 13, 3, 1, 9, 9, 22, 1, 2

Offset: 1

Gus Wiseman, Feb 28 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 section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.

Positions of first appearances are A008578.
The length of this partition is A051903.
The conjugate version is A051904.
For first instead of last part we get A066328.
These partitions are counted by A239455, complement A351293.
Positions of 1 are A360013, complement A381439.
This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).
This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).
Counting partitions by this statistic gives A381438.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[If[n==1,0,Last[egs[prix[n]]]],{n,100}]

a(n) = A055396(A381431(n)).

cp's OEIS Frontend

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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A381990 Number of integer partitions of n that cannot be partitioned into a set (or multiset) of sets with distinct sums.

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A296120 Number of ways to choose a strict factorization of each factor in a strict factorization of n.

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

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

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

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A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

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A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

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A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

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