A383110 Numbers whose prime indices have a permutation with all equal run-sums.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173

Offset: 1

Author

Gus Wiseman, Apr 20 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 prime indices of 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), 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}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}
		

Crossrefs

For distinct run-sums we appear to have complement of A381636 (counted by A381717).

These are the positions of positive terms in A382877.

For run-lengths instead of sums we have complement of A382879, counted by A383013.

For more than one choice we have A383015.

Partitions of this type are counted by A383098.

For a unique choice we have A383099, counted by A383095.

The complement is A383100, counted by A383096.

A056239 adds up prime indices, row sums of A112798.

A304442 counts partitions with equal run-sums, ranks A353833.

A353851 counts compositions with equal run-sums, ranks A353848.

Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.

Cf. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.

Programs

Mathematica

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]>0&]

Formula

Equals A383015 \/ A383099, counted by A353851 \/ A383095.