OFFSET
1,2
COMMENTS
A spanning 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 m is distinct.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Differs from A320058 in having 1155, 1625, 1815, 1875, 1911, ... and lacking 20, 28, 42, 44, 52, ...
EXAMPLE
The sequence of all spanning sum-product knapsack partitions begins: (), (1), (2), (1,1), (3), (2,1), (4), (1,1,1), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (3,3), (6,1).
A complete list of sums of products of multiset partitions of the partition (5,4,3,2) is:
(2*3*4*5) = 120
(2)+(3*4*5) = 62
(3)+(2*4*5) = 43
(4)+(2*3*5) = 34
(5)+(2*3*4) = 29
(2*3)+(4*5) = 26
(2*4)+(3*5) = 23
(2*5)+(3*4) = 22
(2)+(3)+(4*5) = 25
(2)+(4)+(3*5) = 21
(2)+(5)+(3*4) = 19
(3)+(4)+(2*5) = 17
(3)+(5)+(2*4) = 16
(4)+(5)+(2*3) = 15
(2)+(3)+(4)+(5) = 14
These are all distinct, and the Heinz number of (5,4,3,2) is 1155, so 1155 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, facs[#]}]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 04 2018
STATUS
approved