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

1,2

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.

Links

Table of n, a(n) for n=1..45.

Formula

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

Example

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}

Mathematica

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

Crossrefs

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

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

Sequence in context: A166062 A100194 A229882 * A298759 A360680 A369685

Adjacent sequences: A325983 A325984 A325985 * A325987 A325988 A325989

Keyword

nonn

Author

Gus Wiseman, May 30 2019

Status

approved