cp's OEIS Frontend

A320055 Heinz numbers of sum-product knapsack partitions.

%I A320055 #7 Oct 05 2018 18:47:05
%S A320055 1,2,3,5,7,11,13,15,17,19,21,23,25,29,31,33,35,37,39,41,43,47,49,51,
%T A320055 53,55,57,59,61,65,67,69,71,73,77,79,83,85,87,89,91,93,95,97,101,103,
%U A320055 107,109,111,113,115,119,121,123,127,129,131,133,137,139,141,143
%N A320055 Heinz numbers of sum-product knapsack partitions.
%C A320055 A 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 any submultiset of m is distinct.
%C A320055 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A320055 Differs from A320056 in having 2, 845, ... and lacking 245, 455, 847, ....
%e A320055 A complete list of sums of products of multiset partitions of submultisets of the partition (6,6,3) is:
%e A320055             0 = 0
%e A320055           (3) = 3
%e A320055           (6) = 6
%e A320055         (3*6) = 18
%e A320055         (6*6) = 36
%e A320055       (3*6*6) = 108
%e A320055       (3)+(6) = 9
%e A320055     (3)+(6*6) = 39
%e A320055       (6)+(6) = 12
%e A320055     (6)+(3*6) = 24
%e A320055   (3)+(6)+(6) = 15
%e A320055 These are all distinct, and the Heinz number of (6,6,3) is 845, so 845 belongs to the sequence.
%t A320055 multWt[n_]:=If[n==1,1,Times@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]^k]];
%t A320055 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A320055 Select[Range[100],UnsameQ@@Table[Plus@@multWt/@f,{f,Join@@facs/@Divisors[#]}]&]
%Y A320055 Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913.
%Y A320055 Cf. A267597, A320052, A320053, A320054, A320056, A320057, A320058.
%K A320055 nonn
%O A320055 1,2
%A A320055 _Gus Wiseman_, Oct 04 2018