A351295 -id:A351295 - cp's OEIS frontend

A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109

Offset: 1

Gus Wiseman, Apr 18 2025

nonn

First differs from A359178 (complement A362606) in having 1, 240 and lacking 180.
First differs from A130091 (complement A130092) in having 240 and lacking 360.
First differs from A351294 (complement A351295) in having 240 and lacking 216.
Includes all primes A000040 and prime powers A000961.
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 144 are {1,1,1,1,2,2}, with just one permutation with all equal run-lengths (1,1,2,2,1,1), so 144 is in the sequence.
The prime indices of 240 are {1,1,1,1,2,3}, which have no permutation with all equal run-lengths, so 240 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}

These are positions of zeros and ones in A382857, just zeros A382879, just ones A383112.
The complement for run-sums instead of lengths is A383015, counted by A383097.
The complement is A383089, counted by A383090.
Partitions of this type are counted by A383092, just zero A382915, just one A383094.
For run-sums instead of lengths we have A383099 \/ A383100, counted by A383095 + A383096.
A047966 counts partitions with equal run-lengths, compositions A329738.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths, ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A000961, A001221, A001222, A048767, A351294, A353744, A353833, A381435, A382771, A382877, A383113.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]]<=1&]

Equals A382879 \/ A383112, counted by A382915 + A383094.

A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 22, 28, 34, 46, 58, 69, 90, 114, 141, 178, 216, 271, 338, 418, 506, 630, 769, 941, 1140, 1399, 1675, 2051, 2454, 2975, 3561, 4289, 5094, 6137, 7274, 8692, 10269, 12249, 14414, 17128, 20110, 23767, 27872, 32849, 38346, 45094, 52552, 61533

Offset: 0

Gus Wiseman, Apr 19 2025

nonn

			The partition (222211) has 1 permutation with all equal run-lengths: (221122), so is counted under a(10).
The partition (33211111) has no permutation with all equal run-lengths, so is counted under a(13).
The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
The complement is A383090, ranks A383089.
Partitions of this type are ranked by A383091 = positions of terms <= 1 in A382857.
For a unique choice we have A383094, ranks A383112.
For run-sums instead of lengths we have A383095 + A383096, ranks A383099 \/ A383100.
The complement for run-sums is A383097, ranks A383015, positions of terms > 1 in A382877.
Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A006171, A047993, A362608, A381871, A382076, A382771, A383111.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Length/@Split[#]&]]<=1&]],{n,0,15}]

a(n) = A382915(n) + A383094(n).

More terms from Bert Dobbelaere, Apr 26 2025

A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset: 1

Gus Wiseman, May 15 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.

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 terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Partitions of this type are counted by A383507.
Negating both sides gives A383531, counted by A383530.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
paug[y_]:=-DeleteCases[Differences[Append[y,0]],0];
Select[Range[100], UnsameQ@@Last/@FactorInteger[#] && UnsameQ@@paug[Reverse[prix[#]]]&]

Equals A130091 /\ A383512.

A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 21, 27, 30, 33, 41, 50, 57, 68, 79, 89, 112, 126, 144, 172, 198, 220, 257, 298, 327, 383, 423, 477, 533, 621, 650, 760, 816, 920, 1013

Offset: 0

Gus Wiseman, May 19 2025

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.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091).

			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)

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A098859 (A130091), conjugate (A383512).
For non Wilf instead of Wilf we have A383506 (A383514).
These partitions are ranked by (A383520).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
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).
Cf. A047966, A320348, A325324, A325325, A351592, A353837, A381431, A383530, A383709, A384006.

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]}]];
Table[Length[Select[IntegerPartitions[n],disjointFamilies[conj[#]]!={}&&UnsameQ@@Length/@Split[#]&]],{n,0,15}]

A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 06 2025

nonn

First differs from A382525 at a(216) = 1, A382525(216) = 2.

			The prime exponents of 864 = 2^5 * 3^3 are (5,3), with disjoint families {{3},{5}}, {{3},{1,4}}, {{5},{1,2}}, so a(864) = 3.

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
For ordered set partitions we have A382525.
Positions of first appearances are A382775.
The separable case is A386575.
The inseparable case is A386582, see A386632.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A239455 counts Look-and-Say partitions, complement A351293.
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.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A025065, A051903, A051904, A056239, A130091, A336106, A373957, A386580.

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}]]]];
Table[Length[disjointFamilies[prix[n]]],{n,100}]

A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

Original entry on oeis.org

1, 6, 12, 96, 360, 1536, 3456, 5184, 5760, 6144, 7776, 13824, 23040, 24576, 55296, 62208, 92160

Offset: 1

Gus Wiseman, Apr 09 2025

			The permutations for n = 12, 96, 360, 1536:
  (1,1,2)  (1,1,1,1,1,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,1,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,1,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,1,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
                          (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
                          (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
                                         (1,1,2,1,1,1,1,1,1,1)
                                         (2,1,1,1,1,1,1,1,1,1)

Positions of first appearances in A382771, by signature A382773.
For equal run-lengths we have A382878, firsts of A382857, zeros A382879.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A055396 gives least prime index, greatest A061395.
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, A238130, A305936, A351013, A351202, A382876.

y=Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,0,100000}];
fip[y_]:=Select[Range[Length[y]],!MemberQ[Take[y,#-1],y[[#]]]&];
fip[Rest[y]]

A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 20, 23, 25, 28, 29, 31, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 121, 124, 127, 131, 137, 139, 148, 149, 151, 153, 157, 163, 164

Offset: 1

Gus Wiseman, May 15 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.

Integer partitions with distinct multiplicities are called Wilf partitions.

			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}
   13: {6}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   28: {1,1,4}
   29: {10}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   44: {1,1,5}
   45: {2,2,3}
   47: {15}
   49: {4,4}
   50: {1,3,3}

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

For just distinct multiplicities we have A130091 (conjugate A383512), counted by A098859.
For just distinct 0-appended differences we have A325367, counted by A325324.
These partitions are counted by A383709.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383507 counts partitions that are Wilf and conjugate Wilf, ranks A383532.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
Cf. A000720, A005117, A047966, A238745, A320348, A325325, A325349, A325355, A325366, A325368, A325388, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Length/@Split[prix[#]] && UnsameQ@@Differences[Append[Reverse[prix[#]],0]]&]

Equals A130091 /\ A325367.

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&]

A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 216, 224, 232, 236, 242

Offset: 1

Gus Wiseman, Apr 20 2025

nonn

First differs from A177425, A182854, A367589 in having 216.

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 360 are {1,1,1,2,2,3}, with six permutations with all distinct run-lengths:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)
so 360 is in the sequence.
The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  80: {1,1,1,1,3}

For exactly one permutation we have A000961, counted by A000005.
For no choices we have A351293, counted by A351295, conjugate A381433, equal A382879.
For at least one choice we have A351294, conjugate A381432, counted by A239455.
These are positions of terms > 1 in A382771, firsts A382772, equal A382878.
For equal run-lengths we have A383089, positions of terms > 1 in A382857.
Partitions of this type are counted by A383111.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths (ordered A242882), ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A001221, A001222, A047966, A048767, A351013, A351202, A381435, A382876, A383090, A383091, A383092, A383112.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]]>1&]

The complement is A000961 \/ A351293, counted by A000005 + A351295.

cp's OEIS Frontend

A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.

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A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.

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A383532 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct nonzero 0-appended differences (conjugate Wilf).

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A383519 Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

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A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

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A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

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A383712 Heinz numbers of integer partitions with distinct multiplicities (Wilf) and distinct 0-appended differences.

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

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

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