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A301899
Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).
27
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109
OFFSET
1,2
COMMENTS
An integer partition is knapsack if every distinct submultiset has a different sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
FORMULA
Intersection of A299702 and A005117.
EXAMPLE
42 is the Heinz number of (4,2,1) which is strict and knapsack, so is in the sequence. 45 is the Heinz number of (3,2,2) which is knapsack but not strict, so is not in the sequence. 30 is the Heinz number of (3,2,1) which is strict but not knapsack, so is not in the sequence.
Sequence of strict knapsack partitions begins: (), (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (8), (42), (51), (9), (61).
MATHEMATICA
wt[n_]:=If[n===1, 0, Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimePi[p]]]];
Select[Range[100], SquareFreeQ[#]&&UnsameQ@@wt/@Divisors[#]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 28 2018
STATUS
approved