A383100 Numbers whose prime indices have no permutation with all equal run-sums.

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108

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 18 are {1,2,2}, with permutations (1,2,2), (2,1,2), (2,2,1), with run sums (1,4), (2,1,2), (4,1) respectively, so 18 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
   50: {1,3,3}
		

Crossrefs

For distinct instead of equal run-sums we appear to have A381636, counted by A381717.

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

These are the positions of 0 in A382877.

For more than one choice we have A383015.

The complement is A383110, counted by A383098.

Partitions of this type are counted by A383096.

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

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. A351294, A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383094, A383097.

Programs

Mathematica

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