A325284 Numbers whose prime indices form an initial interval with a single hole: (1, 2, ..., x, x + 2, ..., m - 1, m), where x can be 0 but must be less than m - 1.

3, 9, 10, 15, 20, 27, 40, 42, 45, 50, 70, 75, 80, 81, 84, 100, 105, 126, 135, 140, 160, 168, 200, 225, 243, 250, 252, 280, 294, 315, 320, 330, 336, 350, 375, 378, 400, 405, 462, 490, 500, 504, 525, 560, 588, 640, 660, 672, 675, 700, 729, 735, 756, 770, 800

Offset: 1

Author

Gus Wiseman, Apr 19 2019

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose distinct parts form an initial interval with a single hole. The enumeration of these partitions by sum is given by A090858.

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}
    9: {2,2}
   10: {1,3}
   15: {2,3}
   20: {1,1,3}
   27: {2,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   45: {2,2,3}
   50: {1,3,3}
   70: {1,3,4}
   75: {2,3,3}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
  100: {1,1,3,3}
  105: {2,3,4}
  126: {1,2,2,4}
  135: {2,2,2,3}
  140: {1,1,3,4}
		

Crossrefs

Cf. A055932, A056239, A061395, A090858, A112798, A124010, A127002, A130091, A325241, A325251, A325259, A325270.

Programs

Mathematica

Select[Range[100],Length[Complement[Range[PrimePi[FactorInteger[#][[-1,1]]]],PrimePi/@First/@FactorInteger[#]]]==1&]