OFFSET
1,2
COMMENTS
A sum-product knapsack partition is a finite multiset m of positive integers such that every sum of products of parts of any 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 A320056 in having 2, 845, ... and lacking 245, 455, 847, ....
EXAMPLE
A complete list of sums of products of multiset partitions of submultisets of the partition (6,6,3) is:
0 = 0
(3) = 3
(6) = 6
(3*6) = 18
(6*6) = 36
(3*6*6) = 108
(3)+(6) = 9
(3)+(6*6) = 39
(6)+(6) = 12
(6)+(3*6) = 24
(3)+(6)+(6) = 15
These are all distinct, and the Heinz number of (6,6,3) is 845, so 845 belongs to the sequence.
MATHEMATICA
multWt[n_]:=If[n==1, 1, Times@@Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]^k]];
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[Plus@@multWt/@f, {f, Join@@facs/@Divisors[#]}]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 04 2018
STATUS
approved