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A320056 Heinz numbers of product-sum knapsack partitions. 7
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 (list; graph; refs; listen; history; text; internal format)
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: A245644 A070087 A100933 * A175679 A088828 A182318

Adjacent sequences:  A320053 A320054 A320055 * A320057 A320058 A320059

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 04 2018

STATUS

approved

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)