A048768 -id:A048768 - cp's OEIS frontend

A381541 Numbers appearing more than once in A048767 (Look-and-Say partition of prime indices).

8, 16, 27, 32, 64, 81, 96, 125, 128, 144, 160, 192, 216, 224, 243, 256, 288

Offset: 1

Gus Wiseman, Mar 02 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.
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The terms together with their prime indices begin:
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
   96: {1,1,1,1,1,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  160: {1,1,1,1,1,3}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  224: {1,1,1,1,1,4}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
For example, the term 96 appears in A048767 at positions 44 and 60, with prime indices:
  44: {1,1,5}
  60: {1,1,2,3}

In A048767:
- fixed points are A048768, A217605
- conjugate is A381431, fixed points A000961, A000005
- all numbers present are A351294, conjugate A381432
- numbers missing are A351295, conjugate A381433
- numbers appearing only once are A381540, conjugate A381434
- numbers appearing more than once are A381541 (this), conjugate A381435
A000040 lists the primes.
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 partitions, complement A351293.
A381440 lists Look-and-Say partitions of prime indices, conjugate A381436.
Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]>1&]

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

Original entry on oeis.org

325, 845, 931, 1625, 2527, 3509, 6253, 6517, 8125, 9251

Offset: 1

Gus Wiseman, May 18 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.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
A integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its positive 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
   325: {3,3,6}
   845: {3,6,6}
   931: {4,4,8}
  1625: {3,3,3,6}
  2527: {4,8,8}
  3509: {5,5,10}
  6253: {6,6,12}
  6517: {4,4,4,8}
  8125: {3,3,3,3,6}
  9251: {5,10,10}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383511.
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, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A047966, A051903, A212166, A238745, A325368, A351592, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
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]}]];
Select[Range[1000],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Length/@Split[conj[prix[#]]]&]

A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135

Offset: 1

Gus Wiseman, Jun 03 2025

nonn

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

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.

			The prime indices of 102 are {1,2,7}, which has proper disjoint choice ((1),(2),(3,4)), so 102 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           27: {2,2,2}        63: {2,2,4}
     2: {1}          28: {1,1,4}        64: {1,1,1,1,1,1}
     3: {2}          30: {1,2,3}        66: {1,2,5}
     4: {1,1}        32: {1,1,1,1,1}    68: {1,1,7}
     6: {1,2}        36: {1,1,2,2}      70: {1,3,4}
     8: {1,1,1}      40: {1,1,1,3}      72: {1,1,1,2,2}
     9: {2,2}        42: {1,2,4}        75: {2,3,3}
    10: {1,3}        44: {1,1,5}        76: {1,1,8}
    12: {1,1,2}      45: {2,2,3}        78: {1,2,6}
    14: {1,4}        48: {1,1,1,1,2}    80: {1,1,1,1,3}
    15: {2,3}        50: {1,3,3}        81: {2,2,2,2}
    16: {1,1,1,1}    52: {1,1,6}        84: {1,1,2,4}
    18: {1,2,2}      54: {1,2,2,2}      88: {1,1,1,5}
    20: {1,1,3}      56: {1,1,1,4}      90: {1,2,2,3}
    24: {1,1,1,2}    60: {1,1,2,3}      92: {1,1,9}

The non-proper version appears to be A382912, counted by A383710.
The non-proper complement appears to be A382913, counted by A383708.
The complement is A384321, counted by A384317.
These partitions are counted by A384348.
These are the positions of 0 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
A048767 is the Look-and-Say transform, fixed points A048768.
A056239 adds up prime indices, row sums of A112798.
A179009 counts maximally refined strict partitions, ranks A383707.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
Cf. A122111, A130091, A317142, A326080, A351294, A357982, A381454, A382525, A383706, A384320, A384322.

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@@#&];
Select[Range[100],Length[pofprop[prix[#]]]==0&]

A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121

Offset: 1

Gus Wiseman, Mar 02 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 Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}

In A048767:
- fixed points are A048768, A217605
- conjugate is A381431, fixed points A000961, A000005
- all numbers present are A351294, conjugate A381432
- numbers missing are A351295, conjugate A381433
- numbers appearing only once are A381540 (this), conjugate A381434
- numbers appearing more than once are A381541, conjugate A381435
A000040 lists the primes.
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 partitions, complement A351293.
A381440 lists Look-and-Say partition of prime indices, conjugate A381436.
Cf. A000720, A001222, A003557, A047966, A051903, A051904, A066328, A071178, A116861, A130091, A239964.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
hls[y_]:=Product[Prime[Count[y,x]]^x,{x,Union[y]}];
Select[Range[100],Count[hls/@IntegerPartitions[Total[prix[#]]],#]==1&]

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 20, 27, 30, 31, 40, 50, 56, 68, 76, 86, 112, 126, 139, 170, 197, 216, 251, 297, 317, 378, 411, 466, 521, 607, 621, 745, 791, 892, 975, 1123, 1163, 1366, 1439, 1635, 1757, 2021, 2080, 2464, 2599, 2882, 3116, 3572, 3713

Offset: 0

Gus Wiseman, May 14 2025

nonn

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A048768 gives Look-and-Say fixed points, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences.
Cf. A047966, A048767, A111133, A320348, A325324, A325325, A325367, A325368, A325388, A351294, A383506.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#] && UnsameQ@@DeleteCases[Differences[Append[#,0]],0]&]],{n,0,30}]

These partitions have Heinz numbers A130091 /\ A383512.

A384389 Number of proper ways to choose disjoint strict integer partitions of each prime index of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 0, 0, 4, 0, 5, 0, 1, 1, 7, 0, 2, 1, 0, 0, 9, 0, 11, 0, 1, 2, 1, 0, 14, 2, 1, 0, 17, 0, 21, 0, 0, 4, 26, 0, 2, 0, 2, 0, 31, 0, 2, 0, 3, 4, 37, 0, 45, 6, 0, 0, 3, 0, 53, 0, 4, 0, 63, 0, 75, 7, 0, 0, 2, 0, 88, 0, 0, 9

Offset: 1

Gus Wiseman, Jun 01 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, which is disjoint when n is squarefree.

			The prime indices of 65 are {3,6}, and we have proper choices: ((3),(5,1)), ((3),(4,2)), ((2,1),(6)). Hence a(65) = 3.
The prime indices of 175 are {3,3,4}, and we have choices: ((3),(2,1),(4)), ((2,1),(3),(4)), both already proper. Hence a(175) = 2.

Without disjointness we have A357982 - 1, non-strict version A299200 - 1.
This is the proper case of A383706, conjugate version A384005.
Positions of positive terms are A384321.
Positions of 0 are A384349.
Positions of 1 are A384390.
Positions of terms > 1 are A384393.
The conjugate version is A384394.
Positions of first appearances are 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 partitions, ranks A351294
A351293 counts non-Look-and-Say partitions, ranks A351295.
Cf. A382912, A383710, A383711, A382913, A383708, A383533.
Cf. A179009, A279375, A279790, A381454, A382525, A384322, A384347.

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@@#&];
Table[Length[pofprop[prix[n]]],{n,100}]

a(prime(n)) = A000009(n) - 1.

A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

11, 13, 17, 19, 23, 25, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134

Offset: 1

Gus Wiseman, Jun 02 2025

nonn

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

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.

			The prime indices of 275 are {3,3,5}, with a total of 2 proper choices: ((3),(2,1),(5)) and ((2,1),(3),(5)), so 275 is in the sequence.
The terms together with their prime indices begin:
    11: {5}      51: {2,7}      82: {1,13}
    13: {6}      53: {16}       83: {23}
    17: {7}      55: {3,5}      85: {3,7}
    19: {8}      57: {2,8}      86: {1,14}
    23: {9}      58: {1,10}     87: {2,10}
    25: {3,3}    59: {17}       89: {24}
    29: {10}     61: {18}       91: {4,6}
    31: {11}     62: {1,11}     93: {2,11}
    34: {1,7}    65: {3,6}      94: {1,15}
    37: {12}     67: {19}       95: {3,8}
    38: {1,8}    69: {2,9}      97: {25}
    41: {13}     71: {20}      101: {26}
    43: {14}     73: {21}      103: {27}
    46: {1,9}    74: {1,12}    106: {1,16}
    47: {15}     77: {4,5}     107: {28}
    49: {4,4}    79: {22}      109: {29}

Without "proper" we get A384321 (strict A384322), counted by A384317 (strict A384318).
The case of no choices is A384349, counted by A384348.
These are positions of terms > 1 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
Partitions of this type are counted by A384395.
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 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A382912, A383710, A382913, A383708, A383533.
Cf. A179009, A357982, A381454, A382525, A383706, A383707, A384320, A384323.

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@@#&];
Select[Range[100],Length[pofprop[prix[#]]]>1&]

A382775 Least number appearing n times in A048767 (rank of Look-and-Say partition of prime indices).

Original entry on oeis.org

6, 1, 8, 32, 64, 128, 256, 6144, 512, 27648, 1024, 73728, 2048, 147456, 165888, 4096, 248832, 196608, 8192, 497664, 1119744, 393216, 16384, 2239488

Offset: 0

Gus Wiseman, Apr 11 2025

Also the position of first appearance of n in A382525 (number of times n appears in A048767).
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
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 terms together with their prime indices begin:
       6: {1,2}
       1: {}
       8: {1,1,1}
      32: {1,1,1,1,1}
      64: {1,1,1,1,1,1}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
    6144: {1,1,1,1,1,1,1,1,1,1,1,2}
     512: {1,1,1,1,1,1,1,1,1}
   27648: {1,1,1,1,1,1,1,1,1,1,2,2,2}
    1024: {1,1,1,1,1,1,1,1,1,1}
   73728: {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
  147456: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
  165888: {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
  248832: {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}

Positions of first appearances in A382525.
The Look-and-Say partition is ranked by A048767, listed by A381440.
Look-and-Say partitions are counted by A239455, complement A351293.
Look-and-Say partitions are ranked by A351294.
Non-Look-and-Say partitions are ranked by A351295, conjugate A381433.
The section-sum partition is ranked by A381431, listed by A381436.
Section-sum partitions are ranked by A381432.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Cf. A003557, A047966, A048768, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964, A381540.

stp[y_]:=Select[Tuples[Select[IntegerPartitions[#], UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
z=Table[Length[stp[Last/@FactorInteger[n]]],{n,10000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[z,k][[1,1]],{k,0,mnrm[z+1]-1}]

A384010 Heinz numbers of integer partitions such that it is possible to choose a family of disjoint strict partitions, one of each conjugate part.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 27, 30, 32, 36, 48, 54, 60, 64, 72, 81, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192

Offset: 1

Gus Wiseman, May 23 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.

			The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), disjoint family (4,2,1), so 96 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

For multiplicities instead of indices we have A382525.
These partitions are counted by A383708, without ones A383533, complement A383711.
These are the positions of positive terms in A384005.
The complement is A384011, conjugate A383710.
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.
A122111 represent 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.
Cf. A098859, A130091, A279375, A279790, A299200, A357982, A382912, A383706, A383707.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
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}]]]];
Select[Range[100],pof[conj[prix[#]]]!={}&]

A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 33, 81, 183, 402, 856, 1801, 3721, 7646, 15567, 31575

Offset: 0

Gus Wiseman, Jun 05 2025

Conjecture: Also the number of subsets of {1..n} such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each element.

			For the set s = {1,5} we have 5 = 2+3, so s is counted under a(5).
The a(0) = 0 through a(5) = 13 subsets:
  .  .  .  {3}  {3}    {3}
                {4}    {4}
                {2,4}  {5}
                {3,4}  {1,5}
                       {2,4}
                       {2,5}
                       {3,4}
                       {3,5}
                       {4,5}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {3,4,5}

The complement is counted by A326080, allowing repeats A326083.
For strict partitions of n instead of subsets of {1..n} we have A384318, ranks A384322.
First differences are A384391.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A179009 counts maximally refined strict partitions, ranks A383707.
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.
A383706 counts ways to choose disjoint strict partitions of prime indices, non-disjoint A357982, non-strict A299200.
Cf. A279375, A279790, A317141, A317142, A383708, A383710, A384317, A384319, A384320, A384321.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,10}]

cp's OEIS Frontend

A381541 Numbers appearing more than once in A048767 (Look-and-Say partition of prime indices).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A381540 Numbers appearing only once in A048767 (Look-and-Say partition of prime indices).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A383507 Number of Wilf and conjugate Wilf integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A384389 Number of proper ways to choose disjoint strict integer partitions of each prime index of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A382775 Least number appearing n times in A048767 (rank of Look-and-Say partition of prime indices).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A384010 Heinz numbers of integer partitions such that it is possible to choose a family of disjoint strict partitions, one of each conjugate part.

Views

Author

Keywords