OFFSET
1,2
COMMENTS
A product-sum knapsack partition is a finite multiset m of positive integers such that every product of sums of parts of a multiset partition of any submultiset of m is distinct.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Differs from A320055 in having 245, 455, 847, ... and lacking 2, 845, ....
EXAMPLE
A complete list of products of sums of multiset partitions of submultisets of the partition (5,5,4) is:
() = 1
(4) = 4
(5) = 5
(4+5) = 9
(5+5) = 10
(4+5+5) = 14
(4)*(5) = 20
(4)*(5+5) = 40
(5)*(5) = 25
(5)*(4+5) = 45
(4)*(5)*(5) = 100
These are all distinct, and the Heinz number of (5,5,4) is 847, so 847 belongs to the sequence.
MATHEMATICA
heinzWt[n_]:=If[n==1, 0, Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimePi[p]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], UnsameQ@@Table[Times@@heinzWt/@f, {f, Join@@facs/@Divisors[#]}]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 04 2018
STATUS
approved