cp's OEIS Frontend

A320058 Heinz numbers of spanning product-sum knapsack partitions.

%I A320058 #6 Oct 05 2018 18:47:27
%S A320058 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 A320058 34,35,37,38,39,41,42,43,44,46,47,49,50,51,52,53,55,56,57,58,59,61,62,
%U A320058 65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,85,86,87
%N A320058 Heinz numbers of spanning product-sum knapsack partitions.
%C A320058 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 A320058 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A320058 Differs from A320057 in having 20, 28, 42, 44, 52, ... and lacking 1155, 1625, 1815, 1875, 1911, ....
%e A320058 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 A320058 A complete list of products of sums of multiset partitions of the partition (3,1,1) is:
%e A320058       (1+1+3) = 5
%e A320058     (1)*(1+3) = 4
%e A320058     (3)*(1+1) = 6
%e A320058   (1)*(1)*(3) = 3
%e A320058 These are all distinct, and the Heinz number of (3,1,1) is 20, so 20 belongs to the sequence.
%t A320058 heinzWt[n_]:=If[n==1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
%t A320058 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A320058 Select[Range[100],UnsameQ@@Table[Times@@heinzWt/@f,{f,facs[#]}]&]
%Y A320058 Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913.
%Y A320058 Cf. A267597, A320052, A320053, A320054, A320055, A320056, A320057.
%K A320058 nonn
%O A320058 1,2
%A A320058 _Gus Wiseman_, Oct 04 2018