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

1,2

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.

Links

Table of n, a(n) for n=1..65.

Example

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).

Mathematica

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

Crossrefs

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

Sequence in context: A326535 A358977 A284892 * A325467 A325779 A301899

Adjacent sequences: A319312 A319313 A319314 * A319316 A319317 A319318

Keyword

nonn

Author

Gus Wiseman, Sep 17 2018

Status

approved