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

1,2

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, ....

Links

Table of n, a(n) for n=1..61.

Example

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.

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[#]}]&]

Crossrefs

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

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

Sequence in context: A100933 A325128 A352830 * A175679 A088828 A348741

Adjacent sequences: A320053 A320054 A320055 * A320057 A320058 A320059

Keyword

nonn

Author

Gus Wiseman, Oct 04 2018

Status

approved