A382771 -id:A382771 - cp's OEIS frontend

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

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]]

A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 3, 0, 4, 1, 1, 0, 5, 0, 6, 0, 2, 2, 8, 0, 2, 2, 0, 0, 10, 1, 12, 0, 2, 3, 2, 0, 15, 3, 2, 0, 18, 1, 22, 0, 0, 5, 27, 0, 2, 0, 3, 0, 32, 0, 3, 0, 4, 5, 38, 0, 46, 7, 0, 0, 4, 1, 54, 0, 5, 1, 64, 0, 76, 8, 0, 0, 3, 1, 89, 0, 0, 10

Offset: 1

Gus Wiseman, May 15 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 25 are (3,3), for which we have choices ((3),(2,1)) and ((2,1),(3)), so a(25) = 2.
The prime indices of 91 are (4,6), for which we have choices ((4),(6)), ((4),(5,1)), ((4),(3,2,1)), ((3,1),(6)), ((3,1),(4,2)), so a(91) = 5.
The prime indices of 273 are (2,4,6), for which we have choices ((2),(4),(6)), ((2),(4),(5,1)), ((2),(3,1),(6)), so a(273) = 3.

Adding up over all integer partitions gives A279790, strict A279375.
Without disjointness we have A357982, non-strict version A299200.
For multiplicities instead of indices we have A382525.
Positions of 0 appear to be A382912, counted by A383710, odd case A383711.
Positions of positive terms are A382913, counted by A383708, odd case A383533.
Positions of 1 are A383707, counted by A179009.
The conjugate version 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. A098859, A122111, A130091, A209229, A317141, A381454, A382771, A382876.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[pof[prix[n]]],{n,100}]

A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 2, 4, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 1, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 1, 2, 1, 6, 2, 2

Offset: 0

Gus Wiseman, Apr 09 2025

nonn

The first x with a(x) > 1 but A382771(x) > 0 is a(216) = 4, A382771(216) = 4.

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 216 are {1,1,1,2,2,2} and we have permutations:
  (1,1,1,2,2,2)
  (1,2,1,2,1,2)
  (2,1,2,1,2,1)
  (2,2,2,1,1,1)
so a(216) = 4.
The prime indices of 25920 are {1,1,1,1,1,1,2,2,2,2,3} and we have permutations:
  (1,2,1,2,1,2,1,2,1,3,1)
  (1,2,1,2,1,2,1,3,1,2,1)
  (1,2,1,2,1,3,1,2,1,2,1)
  (1,2,1,3,1,2,1,2,1,2,1)
  (1,3,1,2,1,2,1,2,1,2,1)
so a(25920) = 5.

The restriction to signature representatives (A181821) is A382858, distinct A382773.
The restriction to factorials is A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382771.
For run-sums instead of run-lengths we have A382877, distinct A382876.
Positions of first appearances are A382878.
Positions of 0 are A382879.
Positions of terms > 1 are A383089.
Positions of 1 are A383112.
A003963 gives product of prime indices.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary expansion has all equal run-lengths, distinct A328592.
A353744 ranks compositions with equal run-lengths, counted by A329738.
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A238130, A238279, A351201, A351293, A351295.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]], SameQ@@Length/@Split[#]&]],{n,0,100}]

A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 8, 13, 14, 18, 22, 27, 36, 41, 50, 61, 73, 86

Offset: 0

Gus Wiseman, May 07 2025

Also the number of integer partitions y of n whose normal multiset (in which i appears y_i times) is a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (4,2,1)  (7,1)    (8,1)
                                                   (4,3,1)  (4,3,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)

These partitions have Heinz numbers A382913.
Without ones we have A383533, complement A383711.
The number of such families for each Heinz number is A383706.
The complement is counted by A383710, ranks A382912.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
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. A044813, A047966, A089259, A116540, A091602, A130091, A317141, A351013, A381441, A382771, A383013.

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

A382912 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has no permutation with all distinct run-lengths.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Gus Wiseman, Apr 12 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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, prime indices, and corresponding multisets begin:
   4:       {1,1} {1,2}
   8:     {1,1,1} {1,2,3}
   9:       {2,2} {1,1,2,2}
  12:     {1,1,2} {1,1,2,3}
  16:   {1,1,1,1} {1,2,3,4}
  18:     {1,2,2} {1,1,2,2,3}
  20:     {1,1,3} {1,1,1,2,3}
  24:   {1,1,1,2} {1,1,2,3,4}
  27:     {2,2,2} {1,1,2,2,3,3}
  28:     {1,1,4} {1,1,1,1,2,3}
  32: {1,1,1,1,1} {1,2,3,4,5}
  36:   {1,1,2,2} {1,1,2,2,3,4}
  40:   {1,1,1,3} {1,1,1,2,3,4}
  44:     {1,1,5} {1,1,1,1,1,2,3}
  45:     {2,2,3} {1,1,1,2,2,3,3}
  48: {1,1,1,1,2} {1,1,2,3,4,5}
  50:     {1,3,3} {1,1,1,2,2,2,3}
  52:     {1,1,6} {1,1,1,1,1,1,2,3}

The Look-and-Say partition is ranked by A048767, listed by A381440.
Look-and-Say partitions are counted by A239455, ranks A351294.
Non-Look-and-Say partitions are counted by A351293.
For prime indices instead of signature we have A351295, conjugate A381433.
The complement is A382913.
For equal instead of distinct run-lengths we have A382914, see A382858, A382879, A382915.
A056239 adds up prime indices, row sums of A112798.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A381431 lists the section-sum partition of n, ranks A381436, union A381432.
Cf. A000720, A001221, A001222, A181821, A305936, A335125, A351202, A382525, A382771, A382773, A382775, A382857.
Cf. A381636, A381717, A381871, A382876.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{}, Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
Select[Range[100],Not@*lasQ@*nrmptn]

A382913 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has a permutation with all distinct run-lengths.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103

Offset: 1

Gus Wiseman, Apr 12 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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, prime indices, and corresponding multisets begin:
   1:    {} {}
   2:   {1} {1}
   3:   {2} {1,1}
   5:   {3} {1,1,1}
   6: {1,2} {1,1,2}
   7:   {4} {1,1,1,1}
  10: {1,3} {1,1,1,2}
  11:   {5} {1,1,1,1,1}
  13:   {6} {1,1,1,1,1,1}
  14: {1,4} {1,1,1,1,2}
  15: {2,3} {1,1,1,2,2}
  17:   {7} {1,1,1,1,1,1,1}
  19:   {8} {1,1,1,1,1,1,1,1}
  21: {2,4} {1,1,1,1,2,2}
  22: {1,5} {1,1,1,1,1,2}
  23:   {9} {1,1,1,1,1,1,1,1,1}
  25: {3,3} {1,1,1,2,2,2}
  26: {1,6} {1,1,1,1,1,1,2}

Look-and-Say partitions are counted by A239455, ranks A351294.
Non-Look-and-Say partitions are counted by A351293, ranks A351295.
For prime indices instead of signature we have A351294, conjugate A381432.
The Look-and-Say partition of n is listed by A381440, rank A048767.
The complement is A382912.
For equal run-lengths we have the complement of A382914, see A382858, A382879, A382915.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A381431 ranks section-sum partition, listed by A381436.
Cf. A000720, A001221, A001222, A140690, A181821, A305936, A335125, A382525, A382771, A382773, A382775.
Cf. A381636, A381717, A381871, A382876.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&, If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
Select[Range[100],lasQ@*nrmptn]

A382876 Number of ways to permute the prime indices of n so that the run-sums are all different.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 0, 1, 2, 0, 1, 2, 6, 1, 2, 2, 6, 1, 4, 1, 2, 2, 2, 2, 6, 1, 2, 1, 2, 1, 0, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 12 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, sum A056239.

A run in a sequence is a constant consecutive subsequence. The run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums. See A353932 for run-sums of standard compositions.

			For n = 12, none of the permutations (1,1,2), (1,2,1), (2,1,1) has distinct run-sums, so a(12) = 0.
The prime indices of 36 are {1,1,2,2}, and we have permutations: (1,1,2,2), (2,2,1,1), so a(36) = 2.
For n = 90 we have:
  (1,2,2,3)
  (1,3,2,2)
  (2,2,1,3)
  (2,2,3,1)
  (3,1,2,2)
  (3,2,2,1)
So a(90) = 6. The 6 missing permutations are: (1,2,3,2), (2,1,2,3), (2,1,3,2), (2,3,1,2), (2,3,2,1), (3,2,1,2).

Positions of 1 are A000961.
Compositions of this type are counted by A353850, ranked by A353852.
Positions of 0 appear to be A381636, for equal run-sums A383100.
For run-lengths instead of sums we have A382771, equal A382857 (zeros A382879).
For equal instead of distinct run-sums we have A382877.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A001221, A001222, A098859, A130091, A329738, A351013, A351202, A353848, A353851, A354580, A354584.

Table[Length[Select[Permutations[PrimePi /@ Join@@ConstantArray@@@FactorInteger[n]], UnsameQ@@Total/@Split[#]&]],{n,100}]

A382877 Number of ways to permute the prime indices of n so that the run-sums are all equal.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 14 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, sum A056239.

			The a(144) = 4 permutations of {1,1,1,1,2,2} are:
  (1,1,1,1,2,2)
  (1,1,2,1,1,2)
  (2,1,1,2,1,1)
  (2,2,1,1,1,1)
The a(1728) = 4 permutations are:
  (1,1,1,1,1,1,2,2,2)
  (1,1,2,1,1,2,1,1,2)
  (2,1,1,2,1,1,2,1,1)
  (2,2,2,1,1,1,1,1,1)

Compositions of this type are counted by A353851, ranked by A353848.
For run-lengths instead of sums we have A382857 (zeros A382879), distinct A382771.
For distinct instead of equal run-sums we have A382876, counted by A353850.
Positions of terms > 1 are A383015.
Positions of 1 are A383099.
Positions of 0 are A383100 (complement A383110), counted by A383098.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A000961, A001221, A001222, A130091, A329738, A353833, A353852, A354584, A381871.

Table[Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[n]], SameQ@@Total/@Split[#]&]],{n,100}]

A384005 Number of ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 22 2025

nonn

			The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), and we have choices (6,1) and (4,2,1), so a(96) = 2.
The prime indices of 108 are {1,1,2,2,2}, conjugate (5,3), and we have choices (5,3), (5,2,1), (4,3,1), so a(108) = 3.

Adding up over all integer partitions gives A279790, strict A279375.
For multiplicities instead of indices we have conjugate of A382525.
The conjugate version is A383706.
Positive positions are A384010, conjugate A382913, counted by A383708, odd case A383533.
Positions of 0 are A384011.
Without disjointness we have A384179, conjugate A357982, non-strict version A299200.
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. `A098859, A122111, A130091, A179009, A317141, A382771, A383707, A383710.

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}]]]];
Table[Length[pof[conj[prix[n]]]],{n,100}]

a(n) = A383706(A122111(n)).

A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 4, 4, 1, 0, 4, 4, 0, 0, 1, 6, 1, 0, 4, 6, 4, 0, 1, 6, 4, 0, 1, 6, 1, 0, 0, 8, 1, 0, 4, 0, 6, 0, 1, 0, 6, 0, 6, 8, 1, 0, 1, 10, 0, 0, 8, 6, 1, 0, 8, 6, 1, 0, 1, 10, 0, 0, 6, 6, 1, 0, 0, 12, 1, 0, 16

Offset: 1

Gus Wiseman, Apr 09 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(n) partitions for n = 6, 21, 30, 46:
  (1,1,2)  (1,1,1,1,2,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,2,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,2,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,2,1,1,1,1)  (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 1 are A008578.
For anti-run permutations we have A335125.
For just prime indices we have A382771, firsts A382772, equal A382857.
These permutations for factorials are counted by A382774, equal A335407.
For equal instead of distinct run-lengths we have A382858.
Positions of 0 are A382912, complement A382913.
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.
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. A000670, A000720, A003242, A048767, A130091, A181821, A238130, A305936, A351013, A351202, A382876, A382879.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[Select[Permutations[nrmptn[n]],UnsameQ@@Length/@Split[#]&]],{n,100}]

a(n) = A382771(A181821(n)) = A382771(A304660(n)).

cp's OEIS Frontend

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

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A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

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A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

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A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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A382912 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has no permutation with all distinct run-lengths.

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A382913 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has a permutation with all distinct run-lengths.

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A382876 Number of ways to permute the prime indices of n so that the run-sums are all different.

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A382877 Number of ways to permute the prime indices of n so that the run-sums are all equal.

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A384005 Number of ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

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