A383091 Numbers whose prime indices have at most one permutation with all equal run-lengths.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109

Offset: 1

Author

Gus Wiseman, Apr 18 2025

Keywords

• nonn

Comments

First differs from A359178 (complement A362606) in having 1, 240 and lacking 180.

First differs from A130091 (complement A130092) in having 240 and lacking 360.

First differs from A351294 (complement A351295) in having 240 and lacking 216.

Includes all primes A000040 and prime powers A000961.

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 144 are {1,1,1,1,2,2}, with just one permutation with all equal run-lengths (1,1,2,2,1,1), so 144 is in the sequence.
The prime indices of 240 are {1,1,1,1,2,3}, which have no permutation with all equal run-lengths, so 240 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}
  24: {1,1,1,2}
		

Crossrefs

These are positions of zeros and ones in A382857, just zeros A382879, just ones A383112.

The complement for run-sums instead of lengths is A383015, counted by A383097.

The complement is A383089, counted by A383090.

Partitions of this type are counted by A383092, just zero A382915, just one A383094.

For run-sums instead of lengths we have A383099 \/ A383100, counted by A383095 + A383096.

A047966 counts partitions with equal run-lengths, compositions A329738.

A056239 adds up prime indices, row sums of A112798.

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

A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.

Cf. A000720, A000961, A001221, A001222, A048767, A351294, A353744, A353833, A381435, A382771, A382877, A383113.

Programs

Mathematica

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

Formula

Equals A382879 \/ A383112, counted by A382915 + A383094.