A386632 Numbers k such that there is a disjoint inseparable way to choose a strict integer partition of each exponent in the prime factorization of k.

8, 16, 27, 32, 64, 81, 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1331, 1536, 2048, 2187, 2197, 2304, 2401, 2560, 3072, 3125, 3456, 3584, 4096, 4608, 4913, 5120, 5184, 5632, 6144, 6400, 6561, 6656, 6859, 6912, 7168, 8192, 8704, 9216, 9728, 10240, 11264

Offset: 1

Author

Gus Wiseman, Aug 04 2025

Keywords

• nonn

Comments

First cubefull number (A246549) not in this sequence is 216.

The first term that is not a prime power is 1536.

A set partition is inseparable iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.

Examples

			The prime indices of 2304 are {1,1,1,1,1,1,1,1,2,2}, and we have disjoint inseparable choice {{4,3,1},{2}}, so 2304 is in the sequence.
The terms together with their prime indices begin:
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   343: {4,4,4}
   512: {1,1,1,1,1,1,1,1,1}
   625: {3,3,3,3}
   729: {2,2,2,2,2,2}
		

Crossrefs

This is the inseparable case of A351294, positives in A386575, counted by A239455.

Also positions of positive terms in A386582.

A000110 counts set partitions, ordered A000670.

A003242 and A335452 count separations, ranks A333489.

A025065/A386638 counts inseparable type partitions, ranks A335126, sums of A386586.

A325534 counts separable multisets, ranks A335433, sums of A386583.

A325535 counts inseparable multisets, ranks A335448, sums of A386584.

A336106 counts separable type partitions, ranks A335127, sums of A386585.

A386633 counts separable type set partitions, row sums of A386635.

A386634 counts inseparable type set partitions, row sums of A386636.

Cf. A001221, A001222, A051903, A051904, A056239, A130091, A279790, A351293, A373957, A382525, A386587.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dsj[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
insepQ[y_]:=2*Max[y]>Total[y]+1;
Join@@Position[Sign[Table[Length[Select[dsj[prix[n]],insepQ[Length/@#]&]],{n,1000}]],1]