A299200 -id:A299200 - cp's OEIS frontend

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

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

Gus Wiseman, Aug 30 2025

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.

			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.

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.

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@@#&]=={}&]

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

Gus Wiseman, Mar 13 2025

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

			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}

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.

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

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

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

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.

Original entry on oeis.org

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

A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 8, 8, 12, 17, 22, 29, 31, 40, 50, 65, 77, 101, 112, 135, 162, 201

Offset: 0

Gus Wiseman, May 30 2025

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

			For the partition (8,5,2) we have four choices:
  ((8),(4,1),(2))
  ((7,1),(5),(2))
  ((5,3),(4,1),(2))
  ((4,3,1),(5),(2))
Hence (8,5,2) is counted under a(15).
The a(5) = 1 through a(12) = 12 partitions:
  (5)  (6)    (7)  (8)    (9)    (10)     (11)     (12)
       (3,3)       (4,4)  (5,4)  (5,5)    (6,5)    (6,6)
                   (5,3)  (6,3)  (6,4)    (7,4)    (7,5)
                   (7,1)  (7,2)  (7,3)    (8,3)    (8,4)
                          (8,1)  (8,2)    (9,2)    (9,3)
                                 (9,1)    (10,1)   (10,2)
                                 (4,3,3)  (5,3,3)  (11,1)
                                 (4,4,2)  (5,5,1)  (5,5,2)
                                                   (6,3,3)
                                                   (6,4,2)
                                                   (6,5,1)
                                                   (9,2,1)

For just one choice we have A179009, ranked by A383707.
Twice-partitions of this type are counted by A279790.
For at least one choice we have A383708, odd case A383533.
For no choices we have A383710, odd case A383711.
For at least one proper choice we have A384317, ranked by A384321.
The strict version for at least one proper choice is A384318, ranked by A384322.
The strict version for just one proper choice is A384319, ranked by A384390.
For just one proper choice we have A384323, ranks A384347 = positions of 2 in A383706.
For no proper choices we have A384348, ranked by A384349.
These partitions are ranked by A384393.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
A357982 counts choices of strict partitions of each prime index, non-strict A299200.
Cf. A098859, A279375, A317142, A381454, A382525, A382912, A382913, A384005.

pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pofprop[#]]>1&]],{n,0,15}]

A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 15, 8, 9, 14, 56, 12, 101, 30, 21, 16, 297, 18, 490, 28, 45, 112, 1255, 24, 49, 202, 27, 60, 4565, 42, 6842, 32, 168, 594, 105, 36, 21637, 980, 303, 56, 44583, 90, 63261, 224, 63, 2510, 124754, 48, 225, 98, 891, 404, 329931, 54, 392, 120

Offset: 1

Gus Wiseman, Sep 05 2025

			The a(1) = 1 through a(7) = 15 ways:
  (1)  (2)   (3)    (2)(2)    (5)      (2)(3)     (7)
       (11)  (21)   (11)(2)   (32)     (11)(3)    (43)
             (111)  (2)(11)   (41)     (2)(21)    (52)
                    (11)(11)  (221)    (11)(21)   (61)
                              (311)    (2)(111)   (322)
                              (2111)   (11)(111)  (331)
                              (11111)             (421)
                                                  (511)
                                                  (2221)
                                                  (3211)
                                                  (4111)
                                                  (22111)
                                                  (31111)
                                                  (211111)
                                                  (1111111)

For constant partitions we have A061142, for prime indices A355731.
For prime indices instead of factors we have A299200.
The version for distinct choices is A387133, zeros A387326.
A000041 counts integer partitions, strict A000009.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A387110 counts choices of distinct distinct integer partitions of each prime index.
Cf. A063834, A120383, A299201, A335433, A335448, A355741, A357977, A383706, A387115, A387120, A387136.

Table[Length[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]]],{n,30}]

A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 11, 2, 2, 1, 33, 1, 116, 1, 5, 4, 435, 1, 2, 11, 1, 2, 1832, 2, 8167, 1, 12, 33, 10, 1, 39700, 116, 37, 1, 201785, 5, 1099449, 4, 3, 435, 6237505, 1, 19, 2, 123, 11, 37406458, 1, 27, 2, 474, 1832, 232176847, 2, 1513796040

Offset: 1

Gus Wiseman, Mar 04 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(21) = 5 maximal chains are the rows:
(111111)<(21111)<(2211)<(222)<(42)
(111111)<(21111)<(2211)<(411)<(42)
(111111)<(21111)<(2211)<(321)<(42)
(111111)<(21111)<(3111)<(411)<(42)
(111111)<(21111)<(3111)<(321)<(42)

Cf. A000041, A001055, A001222, A002846, A056239, A112798, A213427, A215366, A265947, A296150, A299200, A299202, A299925, A300273, A300383, A300385.

pcovs[ptn_]:=Select[Union[Reverse/@Sort/@Join@@@Tuples[IntegerPartitions/@ptn]],Length[#]===Length[ptn]+1&];
coc[ptn_]:=coc[ptn]=If[Max[ptn]===1,1,Total[coc/@pcovs[ptn]]];
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[coc[Reverse[primeMS[n]]],{n,50}]

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}]

A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 30, 60, 90, 120, 150, 180, 210, 240, 420, 540, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 2520, 3360, 4620, 6300, 6930, 7560, 9240

Offset: 1

Gus Wiseman, Jun 09 2025

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    30: {1,2,3}
    60: {1,1,2,3}
    90: {1,2,2,3}
   120: {1,1,1,2,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   240: {1,1,1,1,2,3}
   420: {1,1,2,3,4}
   540: {1,1,2,2,2,3}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}

Partitions of this type are counted by A179009.
The conjugate version is A383707, proper A384390.
Appears to be the positions of 1 in A384005 (conjugate A383706).
For at least one instead of exactly one choice we appear to have A384010.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A357982 counts strict partitions of prime indices, non-strict A299200.
Cf. A003963, A048767, A130091, A382525, A384317, A384318, A384320, A384347, A384349, A384394.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[100],With[{y=conj[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]

A387326 Numbers whose prime factors do not have choosable sets of integer partitions.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81

Offset: 1

Gus Wiseman, Sep 04 2025

We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

Also numbers n with at least one prime index k such that the multiplicity of k in the prime factors of n exceeds the number of integer partitions of k.

			The prime factors of 72 are {2,2,2,3,3}, with sets of partitions ({(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1),(2)},{(1,1,1),(2,1),(3)},{(1,1,1),(2,1),(3)}), which is not choosable, so 72 is in the sequence.

The version for prime indices differs from A276079 in lacking 16807, counted by A387134.
If we take the set {1..k} instead of the set of integer partitions of k we get A325127.
A subset of A365886.
Positions of zero in A387133.
For prime indices instead of factors we have A387577.
A000041 counts integer partitions, strict A000009.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A387327 counts partitions of prime factors.
A387328 counts partitions with choosable sets of partitions, ranks A387576.
Cf. A063834, A120383, A289509, A299200, A335433, A335448, A355739, A383706, A387111, A387135, A387180.

Select[Range[50],Select[Tuples[IntegerPartitions/@Join@@ConstantArray@@@FactorInteger[#]],UnsameQ@@#&]=={}&]

A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, 1, 2, 0, 7, 0, 11, 0, 0, 1, 15, 0, 22, 0, 0, 0, 30, 0, 3, 0, 2, 0, 42, 0, 56, 1, 0, 0, 0, 0, 77, 0, 0, 0, 101, 0, 135, 0, 0, 0, 176, 0, 5, 0, 0, 0, 231, 0, 0, 0, 0, 0, 297, 0, 385, 0, 0, 1, 0, 0, 490, 0, 0, 0, 627, 0, 792, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 21 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.

If n is a prime power prime(x)^y, then a(n) is the number of integer partitions of x; otherwise, a(n) = 0.

			The a(49) = 5 choices:
  ((4),(4))
  ((3,1),(3,1))
  ((2,2),(2,2))
  ((2,1,1),(2,1,1))
  ((1,1,1,1),(1,1,1,1))

Positions of zeros are A024619, complement A000961.
Twice-partitions of this type are counted by A047968, see also A296122.
For initial intervals instead of partitions we have A055396, see also A387111.
This is the constant case of A299200, see also A357977, A357982.
For disjoint instead of constant we have A383706.
For distinct instead of constant we have A387110.
For divisors instead of partitions we have A387114, see also A355731, A355739.
For strict partitions instead of partitions we have A387117.
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. A008284, A063834, A276078, A335433, A335448, A355529, A355741, A387115, A387120, A387133, A387136.

Table[Switch[n,1,1,?PrimePowerQ,PartitionsP[PrimePi[FactorInteger[n][[1,1]]]],,0],{n,100}]

a(n) = A000041(A297109(n)).

cp's OEIS Frontend

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

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

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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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A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

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A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

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A300384 In the ranked poset of integer partitions ordered by refinement, number of maximal chains from the local minimum to the partition with Heinz number n.

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

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A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.

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A387326 Numbers whose prime factors do not have choosable sets of integer partitions.

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