A323774 -id:A323774 - cp's OEIS frontend

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

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

A323766 Dirichlet convolution of the integer partition numbers A000041 with the number of divisors function A000005.

Original entry on oeis.org

1, 1, 4, 5, 12, 9, 25, 17, 42, 39, 64, 58, 132, 103, 173, 200, 303, 299, 491, 492, 756, 832, 1122, 1257, 1858, 1975, 2646, 3083, 4057, 4567, 6118, 6844, 8913, 10265, 12912, 14931, 19089, 21639, 27003, 31397, 38830, 44585, 55138, 63263, 77371, 89585, 108076

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of constant multiset partitions of constant multiset partitions of integer partitions of n.

			The a(6) = 25 constant multiset partitions of constant multiset partitions of integer partitions of 6:
  ((6))
  ((52))
  ((42))
  ((33))
  ((3)(3))
  ((3))((3))
  ((411))
  ((321))
  ((222))
  ((2)(2)(2))
  ((2))((2))((2))
  ((3111))
  ((2211))
  ((21)(21))
  ((21))((21))
  ((21111))
  ((111111))
  ((111)(111))
  ((11)(11)(11))
  ((111))((111))
  ((11))((11))((11))
  ((1)(1)(1)(1)(1)(1))
  ((1)(1)(1))((1)(1)(1))
  ((1)(1))((1)(1))((1)(1))
  ((1))((1))((1))((1))((1))((1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A000041, A000837, A001970, A034729, A047968, A306017, A319066, A323764, A323765, A323774.

Table[If[n==0,1,Sum[PartitionsP[d]*DivisorSigma[0,n/d],{d,Divisors[n]}]],{n,0,30}]

a(n) = if (n==0, 1, sumdiv(n, d, numbpart(d)*numdiv(n/d))); \\ Michel Marcus, Jan 28 2019

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

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

A323764 Dirichlet self-convolution of the integer partition numbers A000041.

Original entry on oeis.org

1, 1, 4, 6, 14, 14, 34, 30, 64, 69, 112, 112, 228, 202, 330, 394, 575, 594, 956, 980, 1492, 1674, 2228, 2510, 3700, 3965, 5276, 6200, 8126, 9130, 12318, 13684, 17842, 20622, 25808, 29976, 38377, 43274, 53990, 62976, 77912, 89166, 110656, 126522, 154918, 179744

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of multiset partitions of constant multiset partitions of integer partitions of n.

			The a(4) = 14 multiset partitions of constant multiset partitions:
  ((1111))              ((22))      ((4))  ((31))  ((211))
  ((11)(11))            ((2)(2))
  ((11))((11))          ((2))((2))
  ((1)(1)(1)(1))
  ((1))((1)(1)(1))
  ((1)(1))((1)(1))
  ((1))((1))((1)(1))
  ((1))((1))((1))((1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A000837, A001970, A002033, A003238, A034729, A047968, A279787, A305551, A306017, A319056, A323774.

Join[{1},Table[Sum[PartitionsP[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

Original entry on oeis.org

1, 3, 7, 16, 40, 119, 450, 2253, 15207, 139190, 1731703, 29335875, 677864041, 21400069232, 924419728471, 54716596051100, 4443400439075834, 495676372493566749, 76041424515817042402, 16060385520094706930608, 4674665948889147697184915

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.

			The a(1) = 1 through a(4) = 16 partitions of partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (4)(1111)
                             (11111111)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (2)(2)(2)(11)
                             (2)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Seiichi Manyama, Table of n, a(n) for n = 1..120

Cf. A000123, A001970, A002577, A006171, A007716, A034729, A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.

Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]

a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ Michel Marcus, Jan 28 2019

A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 3, 5, 9, 10, 22, 20, 37, 44, 65, 68, 127, 119, 182, 226, 307, 335, 511, 544, 782, 913, 1171, 1359, 1908, 2121, 2738, 3286, 4174, 4821, 6305, 7182, 9108, 10739, 13195, 15548, 19465, 22397, 27477, 32423, 39448, 45843, 55995, 64871, 78343, 91761, 109325

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of strict multiset partitions of constant multiset partitions of integer partitions of n.

			The a(1) = 1 through a(5) = 10 strict multiset partitions of constant multiset partitions of integer partitions:
  ((1))  ((2))     ((3))          ((4))             ((5))
         ((11))    ((21))         ((31))            ((41))
         ((1)(1))  ((111))        ((22))            ((32))
                   ((1)(1)(1))    ((211))           ((311))
                   ((1))((1)(1))  ((1111))          ((221))
                                  ((2)(2))          ((2111))
                                  ((11)(11))        ((11111))
                                  ((1)(1)(1)(1))    ((1)(1)(1)(1)(1))
                                  ((1))((1)(1)(1))  ((1))((1)(1)(1)(1))
                                                    ((1)(1))((1)(1)(1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A000041, A001970, A034729, A047968, A050343, A316980, A319066, A323764, A323766, A323774.

Join[{1}, Table[Sum[PartitionsQ[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A323775 a(n) = Sum_{k = 1...n} k^(2^(n - k)).

Original entry on oeis.org

1, 3, 8, 30, 359, 72385, 4338080222, 18448597098193762732, 340282370354622283774333836315916425069, 115792089237316207213755562747271079374483128445080168204415615259394085515423

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of ways to choose a constant integer partition of each part of a constant integer partition of 2^(n - 1).

			The a(1) = 1 through a(4) = 30 twice-partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (11)(2)       (22)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (1111)(4)
                             (4)(1111)
                             (11111111)
                             (1111)(22)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (11)(2)(2)(2)
                             (2)(11)(2)(2)
                             (2)(2)(11)(2)
                             (2)(2)(2)(11)
                             (11)(11)(2)(2)
                             (11)(2)(11)(2)
                             (11)(2)(2)(11)
                             (2)(11)(11)(2)
                             (2)(11)(2)(11)
                             (2)(2)(11)(11)
                             (11)(11)(11)(2)
                             (11)(11)(2)(11)
                             (11)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Cf. A000123, A001970, A002577, A006171, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323776.

Mathematica
```
Table[Sum[k^2^(n-k),{k,n}],{n,12}]
```

cp's OEIS Frontend

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

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A323766 Dirichlet convolution of the integer partition numbers A000041 with the number of divisors function A000005.

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

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A323764 Dirichlet self-convolution of the integer partition numbers A000041.

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A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

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A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

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A323775 a(n) = Sum_{k = 1...n} k^(2^(n - k)).

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