A355742 -id:A355742 - cp's OEIS frontend

A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

1, 1, 1, 2, 4, 12, 38, 145, 586, 2619, 12096, 58370, 285244, 1436815, 7281062, 37489525, 193417612

Offset: 0

Gus Wiseman, Mar 14 2025

			The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,2,3}    {1,2,3,4}      {1,2,3,4,5}
              {1,1,2,2}  {1,1,2,2,4}    {1,1,2,2,4,5}
                         {1,1,2,3,3}    {1,1,2,3,3,5}
                         {1,1,1,2,2,3}  {1,1,2,3,4,4}
                                        {1,2,2,3,3,4}
                                        {1,1,1,2,2,3,5}
                                        {1,1,1,2,2,4,4}
                                        {1,1,1,2,3,3,4}
                                        {1,1,2,2,2,3,4}
                                        {1,1,2,2,3,3,3}
                                        {1,1,1,1,2,2,3,4}
                                        {1,1,1,2,2,2,3,3}

Set systems: A050342, A116539, A296120, A318361.
The number of possible choices was A152827, non-strict A058694.
Set multipartitions with distinct sums: A279785, A381718.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Constant instead of strict partitions: A381807, A066843.
A000041 counts integer partitions, strict A000009, constant A000005.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@Range[n]]]],{n,0,10}]

a(12)-a(16) from Christian Sievers, Jun 04 2025

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.

A356067 Number of integer partitions of n into relatively prime prime-powers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 11, 7, 18, 16, 26, 27, 43, 41, 65, 65, 92, 100, 137, 142, 194, 210, 270, 295, 379, 410, 519, 571, 699, 782, 947, 1046, 1267, 1414, 1673, 1870, 2213, 2465, 2897, 3230, 3757, 4210, 4871, 5427, 6265, 6997

Offset: 0

Gus Wiseman, Jul 28 2022

nonn

			The a(5) = 1 through a(12) = 7 partitions:
  (32)  .  (43)   (53)   (54)    (73)    (74)     (75)
           (52)   (332)  (72)    (433)   (83)     (543)
           (322)         (432)   (532)   (92)     (552)
                         (522)   (3322)  (443)    (732)
                         (3222)          (533)    (4332)
                                         (542)    (5322)
                                         (722)    (33222)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

This is the relatively prime case of A023894, facs A000688, w/ 1's A023893.
For strict instead of coprime: A054685, facs A050361, with 1's A106244.
The version for factorizations instead of partitions is A354911.
A000041 counts partitions, strict A000009.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A279784 counts twice-partitions where the latter partitions are constant.
A289509 lists numbers whose prime indices are relatively prime.
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A001970, A055887, A063834, A076610, A085970, A355737, A355742.

Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&&GCD@@#==1&]],{n,0,30}]

A357984 Replace prime(k) with A000720(k) in the prime factorization of n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 3, 0, 2, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 0, 1, 0, 4, 0, 5, 0, 3, 0, 4, 0, 5, 0, 3, 0, 6, 0, 6, 0, 2, 0, 6, 0, 4, 0, 4, 0, 6, 0, 6, 0, 4, 0, 7, 0, 7, 0, 2, 0, 6, 0, 8, 0, 4, 0, 8, 0, 8, 0, 4, 0, 6, 0, 8, 0, 1, 0, 9, 0, 8, 0, 4

Offset: 1

Gus Wiseman, Oct 25 2022

			We have 91 = prime(4) * prime(6), so a(91) = pi(4) * pi(6) = 6.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357983.
The version for p instead of pi is A299200, horz A357977, strict A357982.
The version for nu is A355741.
The version for bigomega is A355742.
The horizontal version is A357980.
A000040 lists the prime numbers.
A000720 is PrimePi.
A056239 adds up prime indices, row-sums of A112798.
Cf. A033879, A033880, A063834, A066207, A076610, A296150, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@PrimePi/@primeMS[n],{n,100}]

cp's OEIS Frontend

A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

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

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A356067 Number of integer partitions of n into relatively prime prime-powers.

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A357984 Replace prime(k) with A000720(k) in the prime factorization of n.

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