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%I #7 Oct 05 2018 18:47:13
%S 1,3,5,7,11,13,15,17,19,21,23,25,29,31,33,35,37,39,41,43,47,49,51,53,
%T 55,57,59,61,65,67,69,71,73,77,79,83,85,87,89,91,93,95,97,101,103,107,
%U 109,111,113,115,119,121,123,127,129,131,133,137,139,141,143
%N Heinz numbers of product-sum knapsack partitions.
%C A product-sum knapsack partition is a finite multiset m of positive integers such that every product of sums of parts of a multiset partition of any submultiset of m is distinct.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C Differs from A320055 in having 245, 455, 847, ... and lacking 2, 845, ....
%e A complete list of products of sums of multiset partitions of submultisets of the partition (5,5,4) is:
%e () = 1
%e (4) = 4
%e (5) = 5
%e (4+5) = 9
%e (5+5) = 10
%e (4+5+5) = 14
%e (4)*(5) = 20
%e (4)*(5+5) = 40
%e (5)*(5) = 25
%e (5)*(4+5) = 45
%e (4)*(5)*(5) = 100
%e These are all distinct, and the Heinz number of (5,5,4) is 847, so 847 belongs to the sequence.
%t heinzWt[n_]:=If[n==1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Select[Range[100],UnsameQ@@Table[Times@@heinzWt/@f,{f,Join@@facs/@Divisors[#]}]&]
%Y Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913.
%Y Cf. A267597, A320052, A320053, A320054, A320055, A320057, A320058.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 04 2018
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