%I #6 Oct 05 2018 18:47:27

%S 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,

%T 34,35,37,38,39,41,42,43,44,46,47,49,50,51,52,53,55,56,57,58,59,61,62,

%U 65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,85,86,87

%N Heinz numbers of spanning product-sum knapsack partitions.

%C 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.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%C Differs from A320057 in having 20, 28, 42, 44, 52, ... and lacking 1155, 1625, 1815, 1875, 1911, ....

%e 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).

%e A complete list of products of sums of multiset partitions of the partition (3,1,1) is:

%e (1+1+3) = 5

%e (1)*(1+3) = 4

%e (3)*(1+1) = 6

%e (1)*(1)*(3) = 3

%e These are all distinct, and the Heinz number of (3,1,1) is 20, so 20 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,facs[#]}]&]

%Y Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913.

%Y Cf. A267597, A320052, A320053, A320054, A320055, A320056, A320057.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 04 2018