A351295 -id:A351295 - cp's OEIS frontend

A384392 Number of integer partitions of n whose distinct parts are maximally refined.

1, 1, 2, 2, 4, 6, 7, 10, 14, 20, 24, 33, 41, 55, 70, 88, 110, 140, 171, 214, 265, 324, 397, 485, 588, 711, 861, 1032, 1241, 1486, 1773

Offset: 0

Gus Wiseman, Jun 07 2025

Given any 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 a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (21)   (22)    (32)     (222)     (322)      (332)
       (11)  (111)  (31)    (41)     (321)     (331)      (431)
                    (211)   (221)    (411)     (421)      (521)
                    (1111)  (311)    (2211)    (2221)     (2222)
                            (2111)   (3111)    (3211)     (3221)
                            (11111)  (21111)   (4111)     (3311)
                                     (111111)  (22111)    (4211)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The strict case is A179009, ranks A383707.
For subsets instead of partitions we have A326080, complement A384350.
These partitions are ranked by A384320, complement A384321.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
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.
Cf. A179822, A279375, A279790, A299200, A317142, A326083, A357982, A383706, A383708, A383710, A384317, A384318, A384319, A384391.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@nonsets[#]]=={}&]],{n,0,15}]

A384008 Irregular triangle read by rows where row n lists the first differences of the 0-prepended prime indices of the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 1, 1, 4, 1, 2, 5, 6, 1, 3, 2, 1, 7, 8, 2, 2, 1, 4, 9, 1, 5, 10, 1, 1, 1, 11, 2, 3, 1, 6, 3, 1, 12, 1, 7, 2, 4, 13, 1, 1, 2, 14, 1, 8, 15, 2, 5, 16, 3, 2, 2, 6, 1, 9, 17, 18, 1, 10, 3, 3, 1, 1, 3, 19, 2, 7, 1, 2, 1, 20, 21, 1, 11, 4, 1, 1, 1, 4, 22, 1, 12, 23, 3, 4

Offset: 1

Gus Wiseman, May 23 2025

All rows are different.

			The 28-th squarefree number is 42, with 0-prepended prime indices (0,1,2,4), with differences (1,1,2), so row 28 is (1,1,2).
The squarefree numbers and corresponding rows begin:
    1: ()        23: (9)        47: (15)
    2: (1)       26: (1,5)      51: (2,5)
    3: (2)       29: (10)       53: (16)
    5: (3)       30: (1,1,1)    55: (3,2)
    6: (1,1)     31: (11)       57: (2,6)
    7: (4)       33: (2,3)      58: (1,9)
   10: (1,2)     34: (1,6)      59: (17)
   11: (5)       35: (3,1)      61: (18)
   13: (6)       37: (12)       62: (1,10)
   14: (1,3)     38: (1,7)      65: (3,3)
   15: (2,1)     39: (2,4)      66: (1,1,3)
   17: (7)       41: (13)       67: (19)
   19: (8)       42: (1,1,2)    69: (2,7)
   21: (2,2)     43: (14)       70: (1,2,1)
   22: (1,4)     46: (1,8)      71: (20)

Row-lengths are A072047, sums A243290.
This is the restriction of A383534 (ranked by A383535) to rows of squarefree index.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
A325324 counts partitions with distinct 0-appended differences, ranks A325367.
Cf. A001221, A005117, A055396, A061395, A325325, A355536, A358137, A384009.

sql=Select[Range[100],SquareFreeQ];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Differences[Prepend[prix[sql[[n]]],0]],{n,Length[sql]}]

A384394 Number of proper ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 03 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.

By "proper" we exclude the case of all singletons.

			The prime indices of 216 are {1,1,1,2,2,2}, with conjugate partition (6,3), with proper choices ((6),(2,1)), ((5,1),(3)), and ((4,2),(3)), so a(216) = 3.

Conjugate prime indices are the rows of A122111.
The non-proper version is A384005, conjugate A383706.
This is the conjugate version of A384389 (firsts A384396).
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
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.
See also A382912, counted by A383710, odd case A383711.
See also A382913, counted by A383708, odd case A383533.
Cf. A279790, A357982, A381454, A382525, A384321, A384322, A384347, A384349, A384390, A384393.

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]}]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Table[Length[pofprop[conj[prix[n]]]],{n,100}]

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}]

A384396 Position of first appearance of n in A384389 (proper choices of disjoint strict partitions of each prime index).

Original entry on oeis.org

1, 5, 11, 13, 17, 19, 62, 23, 111, 29, 123, 31, 129, 217, 37, 141, 106, 41, 159, 391, 118, 43

Offset: 0

Gus Wiseman, Jun 03 2025

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.

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

Positions of first appearances in A384389.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294
A351293 counts non-Look-and-Say partitions, ranks A351295.
Cf. A382912, A383710, A382913, A383708.
Cf. A179009, A279790, A357982, A381454, A382525, A383706, A384321, A384322, A384347, A384349, A384390, A384393.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
lv=Table[Length[pofprop[prix[n]]],{n,100}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[lv,x][[1,1]],{x,0,mnrm[lv+1]-1}]

A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 4, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 4, 1, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 4, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 4, 2, 1, 2, 1, 3, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 23 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.

			The prime indices of 180 are {1,1,2,2,3}, conjugate {5,3,1}, and we have choices:
  {{5},{3},{1}}
  {{5},{2,1},{1}}
  {{4,1},{3},{1}}
  {{4,1},{2,1},{1}}
  {{3,2},{3},{1}}
  {{3,2},{2,1},{1}}
so a(180) = 6.

Positions of 1 are A037143, complement A033942.
For multiplicities instead of indices we have A050361.
Adding up over all integer partitions gives A270995, disjoint A279790, strict A279375.
The conjugate version is A357982, disjoint A383706.
The disjoint case is A384005.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
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.
Cf. A122111, A130091, A179009, A382913, A383707, A383710, A384010, A384011.

fop[y_]:=Join@@@Tuples[strptns/@y];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[fop[conj[prix[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]]=={}]&]

cp's OEIS Frontend

A384392 Number of integer partitions of n whose distinct parts are maximally refined.

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A384008 Irregular triangle read by rows where row n lists the first differences of the 0-prepended prime indices of the n-th squarefree number.

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A384394 Number of proper ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

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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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A384396 Position of first appearance of n in A384389 (proper choices of disjoint strict partitions of each prime index).

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A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

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

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