A320055 - OEIS

%I #7 Oct 05 2018 18:47:05

%S 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 53,55,57,59,61,65,67,69,71,73,77,79,83,85,87,89,91,93,95,97,101,103,

%U 107,109,111,113,115,119,121,123,127,129,131,133,137,139,141,143

%N Heinz numbers of sum-product knapsack partitions.

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

%C Differs from A320056 in having 2, 845, ... and lacking 245, 455, 847, ....

%e A complete list of sums of products of multiset partitions of submultisets of the partition (6,6,3) is:

%e             0 = 0

%e           (3) = 3

%e           (6) = 6

%e         (3*6) = 18

%e         (6*6) = 36

%e       (3*6*6) = 108

%e       (3)+(6) = 9

%e     (3)+(6*6) = 39

%e       (6)+(6) = 12

%e     (6)+(3*6) = 24

%e   (3)+(6)+(6) = 15

%e These are all distinct, and the Heinz number of (6,6,3) is 845, so 845 belongs to the sequence.

%t multWt[n_]:=If[n==1,1,Times@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]^k]];

%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[Plus@@multWt/@f,{f,Join@@facs/@Divisors[#]}]&]

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

%Y Cf. A267597, A320052, A320053, A320054, A320056, A320057, A320058.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 04 2018