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A325986
Heinz numbers of complete strict integer partitions.
6
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.
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[#]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 30 2019
STATUS
approved