A383015 Numbers whose prime indices have more than one permutation with all equal run-sums.

12, 40, 63, 112, 144, 325, 351, 352, 675, 832, 931, 1008, 1539, 1600, 1728, 2176, 2875, 3509, 3969, 4864, 6253, 7047, 7056, 8775, 9072, 11776, 12427, 12544, 12691, 16128, 19133, 20736, 20800, 22464, 23125, 26973, 29403, 29696, 32269, 43200, 49392, 57967, 59711

Offset: 1

Author

Gus Wiseman, Apr 14 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.

All terms appear to have even sum of prime indices.

Examples

			The terms together with their prime indices begin:
     12: {1,1,2}
     40: {1,1,1,3}
     63: {2,2,4}
    112: {1,1,1,1,4}
    144: {1,1,1,1,2,2}
    325: {3,3,6}
    351: {2,2,2,6}
    352: {1,1,1,1,1,5}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
    931: {4,4,8}
   1008: {1,1,1,1,2,2,4}
   1539: {2,2,2,2,8}
   1600: {1,1,1,1,1,1,3,3}
   1728: {1,1,1,1,1,1,2,2,2}
		

Crossrefs

Compositions of this type are counted by A353851, ranked by A353848.

Positions of terms > 1 in A382877, zeros A383100 (complement A383014).

For run-lengths instead of sums we have A383089, counted by A383090.

The complement for run-lengths instead of sums is A383091, counted by A383092

Partitions of this type are counted by A383097.

A044813 lists numbers whose binary expansion has distinct run-lengths.

A056239 adds up prime indices, row sums of A112798.

A304442 counts compositions with equal run-sums, complement A382076.

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

A353837 counts partitions with distinct run-sums, ranks A353838.

A353847 gives composition run-sum transformation, for partitions A353832.

A353932 lists run-sums of standard compositions.

Cf. A000720, A000961, A001221, A001222, A329738, A353833, A354584, A381636, A381871, A382857, A382876, A382879.

Programs

Mathematica

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