A351295 -id:A351295 - cp's OEIS frontend

A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

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

Offset: 1

Gus Wiseman, Jul 31 2025

nonn

A set partition is inseparable iff the underlying set has no permutation whose adjacent elements all belong to different blocks. Note that inseparability only depends on the sizes of the blocks.

			The prime indices of 9216 are {1,1,1,1,1,1,1,1,1,1,2,2}, with a(9216) = 2 choices: {{2},{1,4,5}} and {{2},{1,3,6}}. The other 4 disjoint families {{2},{10}}, {{2},{4,6}}, {{2},{3,7}}, {{2},{1,9}} are separable.
The prime indices of 15552 are {1,1,1,1,1,1,2,2,2,2,2}, with a(15552) = 1 choice: {{5},{1,2,3}}. The other 5 disjoint families {{5},{6}}, {{5},{2,4}}, {{6},{2,3}}, {{6},{1,4}}, {{1,5},{2,3}} are separable.

For separable instead of inseparable we have A386575.
This is the inseparable case of A386587 (ordered version A382525).
Positions of positive terms are A386632.
Positions of first appearances are A386637.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A239455 counts Look-and-Say partitions (ranks A351294), complement A351293 (ranks A351295).
A279790 counts disjoint families on strongly normal multisets.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A051903, A051904, A056239, A111133, A130091, A373957, A386580, A386581.

disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
Table[Length[Select[disjointFamilies[prix[n]],seps[Length/@#]=={}&]],{n,100}]

a(2^n) = A111133(n).

A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 6, 0, 0, 0, 96, 0

Offset: 0

Gus Wiseman, Apr 09 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, sum A056239.

			The prime indices of 24 are {1,1,1,2}, with permutations (1,1,1,2) and (2,1,1,1), so a(4) = 2.

For anti-run permutations we have A335407, see also A335125, A382858.
This is the restriction of A382771 to the factorials A000142, equal A382857.
A022559 counts prime indices of n!, sum A081401.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A181821, A351013, A351202, A360015, A382773, A382879.

Table[Length[Select[Permutations[prix[n!]],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

a(n) = A382771(n!).

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}]

A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 8, 9, 13, 17, 26, 27, 43, 51, 61, 78, 103, 115, 153, 174, 213, 255, 316, 354, 442, 508, 610, 701, 848, 950, 1153, 1303, 1539, 1750, 2075, 2318, 2738, 3081

Offset: 0

Gus Wiseman, Apr 20 2025

			The partition (2,1,1) has two permutations with all distinct run-lengths: (1,1,2), (2,1,1), so it is counted under a(4).
The a(4) = 1 through a(9) = 13 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (5111)     (3222)
                          (4111)    (22211)    (6111)
                          (22111)   (41111)    (22221)
                          (31111)   (221111)   (33111)
                          (211111)  (311111)   (51111)
                                    (2111111)  (222111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

For a unique choice we have A000005, ranks A000961.
For at least one choice we have A239455, ranks A351294, conjugate A381432.
For no choices we have A351293, ranks A351295, conjugate A381433.
The complement is A351293 + A000005, ranks too dense.
For equal instead of distinct run-lengths we have A383090, ranks A383089.
These partitions are ranked by A383113 = positions of terms > 1 in A382771.
Cf. A047966, A098859, A130091, A353837, A100471, A100881, A100882, A100883
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A329738 counts compositions with equal run-lengths, ranks A353744.
Cf. A047993, A329739, A381541, A381636, A381717, A382857, A382915, A383013, A383092, A383094, A383097.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], UnsameQ@@Length/@Split[#]&]]>1&]],{n,0,15}]

a(21)-a(38) from Jakub Buczak, May 04 2025

A383535 Heinz number of the positive first differences of the 0-prepended prime indices of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 4, 7, 2, 3, 6, 11, 4, 13, 10, 6, 2, 17, 4, 19, 6, 9, 14, 23, 4, 5, 22, 3, 10, 29, 8, 31, 2, 15, 26, 10, 4, 37, 34, 21, 6, 41, 12, 43, 14, 6, 38, 47, 4, 7, 6, 33, 22, 53, 4, 15, 10, 39, 46, 59, 8, 61, 58, 9, 2, 25, 20, 67, 26, 51, 12, 71, 4, 73

Offset: 1

Gus Wiseman, May 21 2025

nonn

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.

Also Heinz number of the first differences of the distinct 0-prepended prime indices of n.

			The terms together with their prime indices begin:
     1: {}        2: {1}        31: {11}       38: {1,8}
     2: {1}      17: {7}         2: {1}        47: {15}
     3: {2}       4: {1,1}      15: {2,3}       4: {1,1}
     2: {1}      19: {8}        26: {1,6}       7: {4}
     5: {3}       6: {1,2}      10: {1,3}       6: {1,2}
     4: {1,1}     9: {2,2}       4: {1,1}      33: {2,5}
     7: {4}      14: {1,4}      37: {12}       22: {1,5}
     2: {1}      23: {9}        34: {1,7}      53: {16}
     3: {2}       4: {1,1}      21: {2,4}       4: {1,1}
     6: {1,2}     5: {3}         6: {1,2}      15: {2,3}
    11: {5}      22: {1,5}      41: {13}       10: {1,3}
     4: {1,1}     3: {2}        12: {1,1,2}    39: {2,6}
    13: {6}      10: {1,3}      43: {14}       46: {1,9}
    10: {1,3}    29: {10}       14: {1,4}      59: {17}
     6: {1,2}     8: {1,1,1}     6: {1,2}       8: {1,1,1}

For multiplicities instead of differences we have A181819.
Positions of first appearances are A358137.
Positions of squarefree numbers are A383512, counted by A098859.
Positions of nonsquarefree numbers are A383513, counted by A336866.
These are Heinz numbers of rows of A383534.
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. A122111, A124010 (prime signature), A130091, A287352, A325351, A325366, A325368, A355536, A381431, A384008, A384009.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]

A001222(a(n)) = A001221(n).
A056239(a(n)) = A061395(n).
A055396(a(n)) = A055396(n).
A061395(a(n)) = A241919(n).

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}]

A384011 Numbers k such that it is not possible to choose disjoint strict integer partitions of each conjugate prime index of k.

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Gus Wiseman, May 23 2025

nonn

A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798.

			The terms together with their prime indices begin:
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   28: {1,1,4}

The conjugate is A382912.
These complement is counted by A383708, ranks A382913 or A384010.
These partitions are counted by A383710, conjugate A383711.
These are the positions of 0 in A384005, conjugate A383706.
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, A357982, A382525, A383533, 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[#]]]=={}&]

A384348 Number of integer partitions of n with no proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 7, 11, 17, 25, 30, 44, 61, 82, 113, 141, 193, 249, 327, 422, 548, 682, 881, 1106, 1400, 1751

Offset: 0

Gus Wiseman, May 30 2025

nonn

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

			For the partition y = (5,4,2,1) we have the following proper ways to choose strict partitions of each part:
  ((5),(3,1),(2),(1))
  ((4,1),(4,2),(1))
  ((4,1),(3,1),(2),(1))
  ((3,2),(4),(2),(1))
  ((3,2),(3,1),(2),(1))
But none of this is disjoint, so y is counted under a(12).
The a(1) = 1 through a(8) = 17 partitions:
  (1)  (2)   (21)   (22)    (32)     (222)     (322)      (332)
       (11)  (111)  (31)    (41)     (321)     (331)      (422)
                    (211)   (221)    (411)     (421)      (431)
                    (1111)  (311)    (2211)    (511)      (521)
                            (2111)   (3111)    (2221)     (611)
                            (11111)  (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (3311)
                                               (31111)    (4211)
                                               (211111)   (5111)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The strict case is A179009, ranked by A383707.
This is the proper version of A383710, odd case A383711.
This is the proper complement of A383708, odd case A383533.
The complement is counted by A384317, ranks A384321.
The strict version for at least one proper choice is A384318, ranked by A384322.
For just one proper choice we have A384319, ranked by A384390.
For two choices we have A384323, ranks A384347 = positions of 2 in A383706.
These partitions are ranked by A384349.
For more than one proper choice we have A384395, 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.
Cf. A098859, A279790, A299200, A317142, A357982, A381454, A382525, A382912, A382913, A384005.

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

A384391 Number of subsets of {1..n} containing n and some element that is a sum of distinct non-elements.

Original entry on oeis.org

0, 0, 1, 3, 9, 20, 48, 102, 219, 454, 945, 1920, 3925, 7921, 16008

Offset: 0

Gus Wiseman, Jun 06 2025

			The a(0) = 0 through a(6) = 20 subsets:
  .  .  .  {3}  {4}    {5}      {6}
                {2,4}  {1,5}    {1,6}
                {3,4}  {2,5}    {2,6}
                       {3,5}    {3,6}
                       {4,5}    {4,6}
                       {1,4,5}  {5,6}
                       {2,3,5}  {1,3,6}
                       {2,4,5}  {1,5,6}
                       {3,4,5}  {2,3,6}
                                {2,4,6}
                                {2,5,6}
                                {3,4,6}
                                {3,5,6}
                                {4,5,6}
                                {1,3,5,6}
                                {1,4,5,6}
                                {2,3,4,6}
                                {2,3,5,6}
                                {2,4,5,6}
                                {3,4,5,6}

The complement with n is counted by A179822, first differences of A326080.
Partial sums are A384350.
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, A326083, A383708, A383710, A384317, A384318, A384319, A384320, A384321.

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

cp's OEIS Frontend

A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

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A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

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

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A383111 Number of integer partitions of n having more than one permutation with all distinct run-lengths.

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A383535 Heinz number of the positive first differences of the 0-prepended prime indices of n.

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

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

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A384011 Numbers k such that it is not possible to choose disjoint strict integer partitions of each conjugate prime index of k.

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A384348 Number of integer partitions of n with no proper way to choose disjoint strict partitions of each part.

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