A335448 Numbers whose prime indices are inseparable.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352

Offset: 1

Author

Gus Wiseman, Jun 21 2020

Keywords

• nonn

Comments

First differs from A212164 in lacking 72.

First differs from A293243 in lacking 72.

No terms are squarefree.

Also Heinz numbers of inseparable partitions (A325535). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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.

These are also numbers that can be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.

A multiset is inseparable iff its maximal multiplicity is greater than one plus the sum of its remaining multiplicities.

Examples

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
  32: {1,1,1,1,1}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  80: {1,1,1,1,3}
  81: {2,2,2,2}
  88: {1,1,1,5}
  96: {1,1,1,1,1,2}
		

Crossrefs

Complement of A335433.

Separations are counted by A003242 and A335452 and ranked by A333489.

Permutations of prime indices are counted by A008480.

Inseparable partitions are counted by A325535.

Strict permutations of prime indices are counted by A335489.

Cf. A000670, A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Select[Permutations[primeMS[#]],!MatchQ[#,{_,x_,x_,_}]&]=={}&]