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A320058
Heinz numbers of spanning product-sum knapsack partitions.
7
1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87
OFFSET
1,2
COMMENTS
A spanning product-sum knapsack partition is a finite multiset m of positive integers such that every product of sums 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 A320057 in having 20, 28, 42, 44, 52, ... and lacking 1155, 1625, 1815, 1875, 1911, ....
EXAMPLE
The sequence of all spanning product-sum knapsack partitions begins: (), (1), (2), (1,1), (3), (2,1), (4), (1,1,1), (3,1), (5), (6), (4,1), (3,2), (7), (8), (3,1,1), (4,2), (5,1), (9), (3,3), (6,1), (4,1,1).
A complete list of products of sums of multiset partitions of the partition (3,1,1) is:
(1+1+3) = 5
(1)*(1+3) = 4
(3)*(1+1) = 6
(1)*(1)*(3) = 3
These are all distinct, and the Heinz number of (3,1,1) is 20, so 20 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, facs[#]}]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 04 2018
STATUS
approved