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A323091
Number of strict knapsack factorizations of n.
1
1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 4, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 3, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2
OFFSET
1,6
COMMENTS
A strict knapsack factorization is a finite set of positive integers > 1 such that every subset has a different product.
FORMULA
a(prime^n) = A275972(n).
EXAMPLE
The a(144) = 11 factorizations:
(144),
(2*72), (3*48), (4*36),(6*24), (8*18), (9*16),
(2*3*24), (2*4*18), (2*8*9), (3*6*8).
Missing from this list are (2*6*12), (3*4*12), (2*3*4*6), which are not knapsack.
MATHEMATICA
strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[strfacs[n], UnsameQ@@Times@@@Subsets[#]&]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 04 2019
STATUS
approved