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A320057 Heinz numbers of spanning sum-product knapsack partitions.

%I A320057 #6 Oct 05 2018 18:47:20
%S A320057 1,2,3,4,5,6,7,8,10,11,13,14,15,17,19,21,22,23,25,26,29,31,33,34,35,
%T A320057 37,38,39,41,43,46,47,49,50,51,53,55,57,58,59,61,62,65,67,69,71,73,74,
%U A320057 75,77,79,82,83,85,86,87,89,91,93,94,95,97,98,101,103,105
%N A320057 Heinz numbers of spanning sum-product knapsack partitions.
%C A320057 A spanning sum-product knapsack partition is a finite multiset m of positive integers such that every sum of products of parts of any multiset partition of m is distinct.
%C A320057 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A320057 Differs from A320058 in having 1155, 1625, 1815, 1875, 1911, ... and lacking 20, 28, 42, 44, 52, ...
%e A320057 The sequence of all spanning sum-product knapsack partitions begins: (), (1), (2), (1,1), (3), (2,1), (4), (1,1,1), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (3,3), (6,1).
%e A320057 A complete list of sums of products of multiset partitions of the partition (5,4,3,2) is:
%e A320057         (2*3*4*5) = 120
%e A320057       (2)+(3*4*5) = 62
%e A320057       (3)+(2*4*5) = 43
%e A320057       (4)+(2*3*5) = 34
%e A320057       (5)+(2*3*4) = 29
%e A320057       (2*3)+(4*5) = 26
%e A320057       (2*4)+(3*5) = 23
%e A320057       (2*5)+(3*4) = 22
%e A320057     (2)+(3)+(4*5) = 25
%e A320057     (2)+(4)+(3*5) = 21
%e A320057     (2)+(5)+(3*4) = 19
%e A320057     (3)+(4)+(2*5) = 17
%e A320057     (3)+(5)+(2*4) = 16
%e A320057     (4)+(5)+(2*3) = 15
%e A320057   (2)+(3)+(4)+(5) = 14
%e A320057 These are all distinct, and the Heinz number of (5,4,3,2) is 1155, so 1155 belongs to the sequence.
%t A320057 multWt[n_]:=If[n==1,1,Times@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]^k]];
%t A320057 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A320057 Select[Range[100],UnsameQ@@Table[Plus@@multWt/@f,{f,facs[#]}]&]
%Y A320057 Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913.
%Y A320057 Cf. A267597, A320052, A320053, A320054, A320055, A320056, A320058.
%K A320057 nonn
%O A320057 1,2
%A A320057 _Gus Wiseman_, Oct 04 2018