cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-45 of 45 results.

A382304 MM-numbers of multiset partitions into sets with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 13, 16, 17, 25, 27, 29, 31, 32, 41, 43, 47, 59, 64, 67, 73, 79, 81, 83, 101, 109, 113, 121, 125, 127, 128, 137, 139, 143, 149, 157, 163, 167, 169, 179, 181, 191, 199, 211, 233, 241, 243, 256, 257, 269, 271, 277, 283, 289, 293, 313, 317
Offset: 1

Views

Author

Gus Wiseman, Apr 01 2025

Keywords

Comments

Also products of prime numbers of squarefree index with a common sum of prime indices.
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 multiset of multisets with MM-number n is 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}}.

Examples

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}

Links

Crossrefs

Set partitions of this type are counted by A035470.
Twice-partitions of this type are counted by A279788.
For just strict blocks we have A302478.
For just a common sum we have A326534, distinct sums A326535.
Factorizations of this type are counted by A382080.
For distinct instead of equal sums we have A382201.
For constant instead of strict blocks we have A382215.
Normal multiset partitions of this type are counted by A382429.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A058891 counts set-systems, covering A003465, connected A323818.
Cf. A000720, A005117, A050320, A050326, A293511, A302242, A302494, A381635, A381636, A381995.

Programs

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

Formula

Equals A302478 /\ A326534.

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

Views

Author

Gus Wiseman, Mar 26 2025

Keywords

Comments

Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.

Examples

			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)

Crossrefs

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.

Programs

  • Mathematica
    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}]

A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 2, 4, 12, 24, 92, 184, 704, 2016, 7600, 15200, 80664, 161328, 601696, 2198824, 9868544, 19737088, 102010480, 204020960
Offset: 0

Views

Author

Gus Wiseman, Mar 13 2025

Keywords

Comments

A constant partition is a multiset whose parts are all equal. There are A000005(n) constant partitions of n.

Examples

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

Crossrefs

The number of possible choices was A066843.
Multiset partitions into constant blocks: A006171, A279784, A295935.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Sets of constant multisets with distinct sums: A381635, A381636, A381716.
Strict instead of constant partitions: A381808, A058694, A152827.
A000041 counts integer partitions, strict A000009, constant A000005.
A000688 counts multiset partitions into constant blocks.
A050361 and A381715 count multiset partitions into constant multisets.
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.

Programs

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

Formula

Primorial case of A381453: a(n) = A381453(A002110(n)).

Extensions

a(16)-a(19) from Christian Sievers, Jun 04 2025

A381872 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 having a common sum.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Mar 14 2025

Keywords

Comments

First differs from A321455 at a(144) = 4, A321455(144) = 3.
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.

Examples

			The prime indices of 144 are {1,1,1,1,2,2}, with the following 4 multiset partitions having common block sum:
  {{1,1,1,1,2,2}}
  {{2,2},{1,1,1,1}}
  {{1,1,2},{1,1,2}}
  {{2},{2},{1,1},{1,1}}
with sums: 8, 4, 4, 2, of which 3 are distinct, so a(144) = 3.
The prime indices of 1296 are {1,1,1,1,2,2,2,2}, with the following 7 multiset partitions having common block sum:
  {{1,1,1,1,2,2,2,2}}
  {{2,2,2},{1,1,1,1,2}}
  {{1,1,2,2},{1,1,2,2}}
  {{2,2},{2,2},{1,1,1,1}}
  {{2,2},{1,1,2},{1,1,2}}
  {{1,2},{1,2},{1,2},{1,2}}
  {{2},{2},{2},{2},{1,1},{1,1}}
with sums: 12, 6, 6, 4, 4, 3, 2, of which 5 are distinct, so a(1296) = 5.

Crossrefs

With equal blocks instead of sums we have A089723.
Without equal sums we have A317141, before sums A001055, lower A300383.
Positions of terms > 1 are A321454.
Before taking sums we had A321455.
With distinct instead of equal sums we have A381637, before sums A321469.
A000041 counts integer partitions, strict A000009, constant A000005.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For sets of constant multisets (A050361) see A381715.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
Cf. A000720, A000961, A001222, A279787, A279789, A305551, A306017, A321451, A321452, A321453.

Programs

  • Mathematica
    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]],SameQ@@Total/@#&]]],{n,100}]

A382426 MM-numbers of sets of constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 38, 41, 42, 46, 51, 53, 55, 57, 59, 62, 66, 67, 69, 77, 82, 83, 85, 93, 95, 97, 102, 103, 106, 109, 110, 114, 115, 118, 119, 123, 127, 131, 133, 134, 138, 154, 155, 157, 159, 161, 165, 166
Offset: 1

Views

Author

Gus Wiseman, Apr 01 2025

Keywords

Comments

Also products of prime numbers of prime power index with distinct sums of prime indices.
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 multiset of multisets with MM-number n is 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}}.

Examples

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  30: {{},{1},{2}}

Crossrefs

Twice-partitions of this type are counted by A279786.
For just constant blocks we have A302492.
For just distinct sums we have A326535.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A382201.
Normal multiset partitions of this type are counted by A382203.
For equal instead of distinct sums we have A382215.
An opposite version is A382304.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A000961, A302242, A302494, A321469, A326534, A355743, A356065, A381636, A381716.

Programs

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

Formula

Equals A302492 /\ A326535.
Previous Showing 41-45 of 45 results.

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