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.

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A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

Original entry on oeis.org

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204
Offset: 1

Views

Author

Gus Wiseman, Aug 30 2025

Keywords

Comments

First differs from A276079 in having 125.
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.
Also numbers n with at least one prime index k such that the multiplicity of prime(k) in the prime factorization of n exceeds the number of divisors of k.

Examples

			The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
  ((1),(1),(2),(3))
  ((1),(1),(2),(1,1,1))
  ((1),(1),(1,1),(3))
  ((1),(1),(1,1),(1,1,1))
Since none of these is strict, 60 is in the sequence.
The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
  ((1),(2),(1,1),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(3))
  ((1),(1,1),(2),(1,1,1))
So 90 is not in the sequence.

Crossrefs

For prime factors instead of constant partitions we have A355529, counted by A370593.
For divisors instead of constant partitions we have A355740, counted by A370320.
The complement for prime factors is A368100, counted by A370592.
The complement for divisors is A368110, counted by A239312.
The complement for initial intervals is A387112, counted by A238873.
For initial intervals instead of partitions we have A387113, counted by A387118.
These are the positions of zero in A387120.
For strict instead of constant partitions we have A387176, counted by A387137.
The complement for strict partitions is A387177, counted by A387178.
Twice-partitions of this type are counted by A387179, constant-block case of A296122.
The complement is A387181 (nonzeros of A387120), counted by A387330.
Partitions of this type are counted by A387329.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A355739, A367771, A387111, A387115.
Cf. A000005, A052335, A063834, A276079, A299200, A299201, A335433, A335448, A355731, A383706, A387110.

Programs

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

A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 62, 165, 475, 1400, 4334
Offset: 0

Views

Author

Gus Wiseman, Mar 12 2024

Keywords

Comments

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements.

Examples

			The multiset partition {{3},{1,3},{2,3}} has unique choice (3,1,2) so is counted under a(5).
Representatives of the a(1) = 1 through a(5) = 23 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}
       {1}{2}  {1}{22}    {1}{122}      {11}{122}
               {2}{12}    {11}{22}      {1}{1222}
               {1}{2}{3}  {12}{12}      {11}{222}
                          {1}{222}      {12}{122}
                          {12}{22}      {1}{2222}
                          {2}{122}      {12}{222}
                          {1}{2}{33}    {2}{1122}
                          {1}{3}{23}    {2}{1222}
                          {1}{2}{3}{4}  {22}{122}
                                        {1}{2}{233}
                                        {1}{22}{33}
                                        {1}{23}{23}
                                        {1}{2}{333}
                                        {1}{23}{33}
                                        {1}{3}{233}
                                        {2}{12}{33}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {1}{2}{3}{44}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

Crossrefs

For existence we have A368098, complement A368097.
Multisets of this type are ranked by A368101, see also A368100, A355529.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594, see also A370592, A370593.
Subsets of this type are also counted by A370638, see also A370636, A370637.
Factorizations of this type are A370645, see also A368414, A368413.
Set-systems of this type are A370818, see also A367902, A367903.
A000110 counts set partitions, non-isomorphic A000041.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000612, A055621, A283877, A300913, A302545, A316983, A319616, A330223, A368095, A368412, A368422.

A387328 Number of integer partitions of n whose parts have choosable sets of integer partitions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 17, 22, 28, 36, 46, 58, 73, 91, 114, 141, 174, 214, 262, 320, 389, 472, 571, 688, 828, 993, 1189, 1419, 1690, 2009, 2383, 2821, 3334, 3931, 4628, 5439, 6381, 7474, 8741, 10207, 11902, 13858, 16114, 18710, 21698, 25130, 29070
Offset: 0

Views

Author

Gus Wiseman, Sep 01 2025

Keywords

Comments

First differs from A052335 at A052335(20) = 173, a(20) = 174, corresponding to the partition (4,4,4,4,4).
a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct integer partitions, one of each part.
Also the number of integer partitions y of n with no part k whose multiplicity in y exceeds A000041(k).

Examples

			The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
            (21)  (22)  (32)   (33)   (43)   (44)    (54)
                  (31)  (41)   (42)   (52)   (53)    (63)
                        (221)  (51)   (61)   (62)    (72)
                               (321)  (322)  (71)    (81)
                                      (331)  (332)   (333)
                                      (421)  (422)   (432)
                                             (431)   (441)
                                             (521)   (522)
                                             (3221)  (531)
                                                     (621)
                                                     (3321)
                                                     (4221)

Crossrefs

The strict case is A000009.
For initial intervals instead of partitions we have A238873, complement A387118.
For divisors instead of partitions we have A239312, complement A370320.
For prime factors instead of partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement is counted by A387134, ranks A387577.
For sets of strict partitions we have A387178, complement A387137.
These partitions are ranked by A387576.
A000005 counts divisors.
A000041 counts integer partitions.
Cf. A052335, A052337, A335433, A367771, A370585, A370804.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]!={}&]],{n,0,15}]
Previous Showing 41-43 of 43 results.

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