A325767 Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

1, 2, 12, 18, 36, 60, 120, 180, 360, 450, 540, 600, 840, 1260, 1350, 1500, 1680, 1800, 2250, 2520, 2700, 3000, 3780, 4200, 4500, 5040, 5400, 5880, 6750, 8400, 9000, 10500, 11340, 11760, 12600, 13500, 15120, 17640, 18480, 18900, 20580, 21000, 22680, 25200

Offset: 1

Author

Gus Wiseman, May 19 2019

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A325766.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   180: {1,1,2,2,3}
   360: {1,1,1,2,2,3}
   450: {1,2,2,3,3}
   540: {1,1,2,2,2,3}
   600: {1,1,1,2,3,3}
   840: {1,1,1,2,3,4}
  1260: {1,1,2,2,3,4}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  1680: {1,1,1,1,2,3,4}
  1800: {1,1,1,2,2,3,3}
  2250: {1,2,2,3,3,3}
  2520: {1,1,1,2,2,3,4}
		

Crossrefs

Cf. A000009, A055932, A056239, A112798, A109297, A114640, A290689, A325702, A325708, A325766.

Programs

Mathematica

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[1000],#==1||Range[PrimeNu[#]]==PrimePi/@First/@FactorInteger[#]&&Divisible[#,red[#]]&]