A320056 Heinz numbers of product-sum knapsack partitions.

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 143

Offset: 1

Author

Gus Wiseman, Oct 04 2018

Keywords

• nonn

Comments

A product-sum knapsack partition is a finite multiset m of positive integers such that every product of sums of parts of a multiset partition of any submultiset of m is distinct.

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

Differs from A320055 in having 245, 455, 847, ... and lacking 2, 845, ....

Examples

			A complete list of products of sums of multiset partitions of submultisets of the partition (5,5,4) is:
           () = 1
          (4) = 4
          (5) = 5
        (4+5) = 9
        (5+5) = 10
      (4+5+5) = 14
      (4)*(5) = 20
    (4)*(5+5) = 40
      (5)*(5) = 25
    (5)*(4+5) = 45
  (4)*(5)*(5) = 100
These are all distinct, and the Heinz number of (5,5,4) is 847, so 847 belongs to the sequence.
		

Crossrefs

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

Cf. A267597, A320052, A320053, A320054, A320055, A320057, A320058.

Programs

Mathematica

heinzWt[n_]:=If[n==1,0,Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],UnsameQ@@Table[Times@@heinzWt/@f,{f,Join@@facs/@Divisors[#]}]&]