A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

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 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}
   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}
   41: {13}
		

Crossrefs

For distinct instead of equal run-sums we have A000961, counted by A000005.

These are the positions of 1 in A382877.

For more than one choice we have A383015.

Partitions of this type are counted by A383095.

For no choices we have A383100, counted by A383096.

For at least one choice we have A383110, counted by A383098, see A383013.

For run-lengths instead of sums we have A383112 = positions of 1 in A382857.

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. A000720, A001221, A001222, A351294, A353832, A353838, A353932, A354584, A382876, A383091.

Programs

Mathematica

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

Formula

The complement is A383015 \/ A383100, for run-lengths A382879 \/ A383089.