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

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

Author

Gus Wiseman, Apr 09 2025

Keywords

• nonn

Comments

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.

Examples

			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.
		

Crossrefs

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.

Programs

Mathematica

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