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%I #7 Apr 08 2018 20:09:46
%S 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,
%T 41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,71,73,74,77,78,79,
%U 82,83,85,86,87,89,91,93,94,95,97,101,102,103,105,106,107,109
%N Heinz numbers of strict knapsack partitions. Squarefree numbers such that every divisor has a different Heinz weight A056239(d).
%C 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).
%F Intersection of A299702 and A005117.
%e 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.
%e Sequence of strict knapsack partitions begins: (), (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (8), (42), (51), (9), (61).
%t wt[n_]:=If[n===1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
%t Select[Range[100],SquareFreeQ[#]&&UnsameQ@@wt/@Divisors[#]&]
%Y Cf. A000712, A005117, A056239, A108917, A112798, A122768, A275972, A276024, A284640, A296150, A299701, A299702, A299729, A301829, A301854, A301900.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 28 2018
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