A381993 -id:A381993 - cp's OEIS frontend

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

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!

A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 17, 19, 23, 25, 27, 31, 35, 41, 49, 53, 59, 67, 81, 83, 97, 103, 109, 121, 125, 127, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461, 509, 529, 547, 563, 587, 599

Offset: 1

Gus Wiseman, Apr 25 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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The systems with these MM-numbers begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}
  67: {{8}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  97: {{3,3}}

Twice-partitions of this type are counted by A279789.
For just a common sum we have A326534.
For just constant blocks we have A355743.
Numbers without a factorization of this type are listed by A381871, counted by A381993.
The multiplicative version is A381995.
This is the odd case of A382215.
For strict instead of constant blocks we have A382304.
A001055 counts factorizations, strict A045778.
A023894 counts partitions into prime-powers.
A034699 gives maximal prime-power divisor.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A317141 counts coarsenings of prime indices, refinements A300383.
A353864 counts rucksack partitions, ranked by A353866.
A355742 chooses a prime-power divisor of each prime index.
Cf. A000688, A000720, A001222, A006171, A038041, A279784, A302242, A302493, A321455, A326518, A381719.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Total/@prix/@prix[#]&&And@@PrimePowerQ/@prix[#]&]

Equals A326534 /\ A355743.

A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 10, 13, 15, 13, 31

Offset: 0

Gus Wiseman, Apr 25 2025

Any strict partition can be partitioned into a single set, so we have a lower bound a(n) >= A000009(n).

			The multiset (3,2,2,1,1) has partition {{3},{1,2},{1,2}}, so is counted under a(9).
The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)        (54)
             (111)  (31)    (41)     (42)      (52)       (53)        (63)
                    (1111)  (11111)  (51)      (61)       (62)        (72)
                                     (222)     (421)      (71)        (81)
                                     (321)     (1111111)  (431)       (333)
                                     (2211)               (521)       (432)
                                     (111111)             (2222)      (531)
                                                          (3311)      (621)
                                                          (11111111)  (3321)
                                                                      (32211)
                                                                      (222111)
                                                                      (111111111)

Twice-partitions of this type (into sets with a common sum) are counted by A279788.
Multiset partitions of this type are ranked by A326534 /\ A302478.
For distinct instead of equal sums we have A381992, see also A382077.
The complement is counted by A381994, ranks A381719.
Partitions of prime indices of this type are counted by A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
For constant instead of strict blocks we have A383093, ranks A383014.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A050320, A293511, A321455, A358914, A381633, A381717, A381993.

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[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&SameQ@@Total/@#&]]>0&]],{n,0,10}]

cp's OEIS Frontend

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

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A383309 Numbers whose prime indices are prime powers > 1 with a common sum of prime indices.

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A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

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Mathematica