A320057 Heinz numbers of spanning sum-product knapsack partitions.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 101, 103, 105

Offset: 1

Author

Gus Wiseman, Oct 04 2018

Keywords

• nonn

Comments

A spanning 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 m is distinct.

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

Differs from A320058 in having 1155, 1625, 1815, 1875, 1911, ... and lacking 20, 28, 42, 44, 52, ...

Examples

			The sequence of all spanning sum-product knapsack partitions begins: (), (1), (2), (1,1), (3), (2,1), (4), (1,1,1), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (3,3), (6,1).
A complete list of sums of products of multiset partitions of the partition (5,4,3,2) is:
        (2*3*4*5) = 120
      (2)+(3*4*5) = 62
      (3)+(2*4*5) = 43
      (4)+(2*3*5) = 34
      (5)+(2*3*4) = 29
      (2*3)+(4*5) = 26
      (2*4)+(3*5) = 23
      (2*5)+(3*4) = 22
    (2)+(3)+(4*5) = 25
    (2)+(4)+(3*5) = 21
    (2)+(5)+(3*4) = 19
    (3)+(4)+(2*5) = 17
    (3)+(5)+(2*4) = 16
    (4)+(5)+(2*3) = 15
  (2)+(3)+(4)+(5) = 14
These are all distinct, and the Heinz number of (5,4,3,2) is 1155, so 1155 belongs to the sequence.
		

Crossrefs

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

Cf. A267597, A320052, A320053, A320054, A320055, A320056, A320058.

Programs

Mathematica

multWt[n_]:=If[n==1,1,Times@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]^k]];
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[Plus@@multWt/@f,{f,facs[#]}]&]