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

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

Author

Gus Wiseman, Apr 20 2025

Keywords

• nonn

Comments

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.

Examples

			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}
		

Crossrefs

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.

Programs

Mathematica

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

Formula

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