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

1,2

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, ...

Links

Table of n, a(n) for n=1..66.

Example

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.

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[#]}]&]

Crossrefs

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

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

Sequence in context: A356841 A181709 A320058 * A365406 A342081 A033106

Adjacent sequences: A320054 A320055 A320056 * A320058 A320059 A320060

Keyword

nonn

Author

Gus Wiseman, Oct 04 2018

Status

approved