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

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

Author

Gus Wiseman, Apr 14 2025

Keywords

• nonn

Comments

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 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)
		

Crossrefs

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.

Programs

Mathematica

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