A383112 Numbers whose multiset of prime indices has exactly one permutation with all equal run-lengths.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 81, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 113, 116, 117, 121, 124, 125, 127

Offset: 1

Author

Gus Wiseman, Apr 18 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.

Includes all prime powers A000961.

Are there any terms x such that A001221(x) > 2?

Examples

			The prime indices of 144 are {1,1,1,1,2,2}, of which the only permutation with all equal run-lengths is (1,1,2,2,1,1), so 144 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}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
		

Crossrefs

These are the positions of 1 in A382857, distinct A382771.

The complement is A382879 \/ A383089, counted by A382915 + A383090.

For at most one permutation we have A383091, counted by A383092.

Partitions of this type are counted by A383094.

For run-sums instead of lengths we have A383099, counted by A383095.

A047966 counts partitions with equal run-lengths, ranks A072774.

A056239 adds up prime indices, row sums of A112798.

A098859 counts partitions with distinct run-lengths, ranks A130091.

A329738 counts compositions with equal run-lengths, ranks A353744.

A329739 counts compositions with distinct run-lengths, ranks A351596.

Cf. A000961, A001221, A001222, A048767, A351294, A351295, A353833, A381434, A381540, A382877, A383100.

Programs

Mathematica

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