A319315 Heinz numbers of integer partitions such that every distinct submultiset has a different average.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107

Offset: 1

Author

Gus Wiseman, Sep 17 2018

Keywords

• nonn

Comments

Note that such a Heinz number is necessarily squarefree, as such a partition is necessarily strict.

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

First differs from A301899 at a(43) = 70, because (4,3,1) is not knapsack but every submultiset has a different average.

Examples

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (), (1), (2), (3), (2,1), (4), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (6,1), (10), (11), (5,2), (7,1), (4,3), (12), (8,1), (6,2), (13), (4,2,1).
		

Crossrefs

Cf. A000009, A056239, A122768, A237984, A275972, A299702, A301899, A316313, A316314, A316465.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Mean/@Union[Subsets[primeMS[#]]]&]