A325986 Heinz numbers of complete strict integer partitions.

1, 2, 6, 30, 42, 210, 330, 390, 462, 510, 546, 714, 798, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7854, 8778, 8970, 9282, 9570, 9690, 10230, 10374, 10626, 11310, 11730, 12090, 12210, 12558, 13398, 13566, 14322, 14430

Offset: 1

Author

Gus Wiseman, May 30 2019

Keywords

• nonn

Comments

Strict partitions are counted by A000009, while complete partitions are counted by A126796.

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

An integer partition of n is complete (A126796, A325781) if every number from 0 to n is the sum of some submultiset of the parts.

The enumeration of these partitions by sum is given by A188431.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     42: {1,2,4}
    210: {1,2,3,4}
    330: {1,2,3,5}
    390: {1,2,3,6}
    462: {1,2,4,5}
    510: {1,2,3,7}
    546: {1,2,4,6}
    714: {1,2,4,7}
    798: {1,2,4,8}
   2310: {1,2,3,4,5}
   2730: {1,2,3,4,6}
   3570: {1,2,3,4,7}
   3990: {1,2,3,4,8}
   4290: {1,2,3,5,6}
   4830: {1,2,3,4,9}
   5610: {1,2,3,5,7}
		

Crossrefs

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A304793.

Cf. A325780, A325781, A325782, A325788, A325790, A325988.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Select[Range[1000],SquareFreeQ[#]&&Union[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

Formula

Intersection of A005117 (strict partitions) and A325781 (complete partitions).