A320054 -id:A320054 - cp's OEIS frontend

A267597 Number of sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of any submultiset of y is distinct.

1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 6, 7, 8, 12, 12, 14, 18, 23, 23, 32, 30, 35, 50, 48, 47, 56, 80, 77, 87, 105, 100, 134, 139, 145, 194, 170, 192, 250

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of product-sum knapsack partitions begins:
   0: ()
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5) (3,2)
   6: (6) (4,2) (3,3)
   7: (7) (5,2) (4,3)
   8: (8) (6,2) (5,3) (4,4)
   9: (9) (7,2) (6,3) (5,4)
  10: (10) (8,2) (7,3) (6,4) (5,5) (4,3,3)
  11: (11) (9,2) (8,3) (7,4) (6,5) (5,4,2) (5,3,3)
The partition (4,4,3) is not a sum-product knapsack partition of 11 because (4*4) = (4)+(4*3).
A complete list of all sums of products of multiset partitions of submultisets of (5,4,2) is:
            0 = 0
          (2) = 2
          (4) = 4
          (5) = 5
        (2*4) = 8
        (2*5) = 10
        (4*5) = 20
      (2*4*5) = 40
      (2)+(4) = 6
      (2)+(5) = 7
    (2)+(4*5) = 22
      (4)+(5) = 9
    (4)+(2*5) = 14
    (5)+(2*4) = 13
  (2)+(4)+(5) = 11
These are all distinct, so (5,4,2) is a sum-product knapsack partition of 11.

Sean A. Irvine, Java program (github)

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A320052, A320053, A320054, A320055, A320056, A320057, A320058.

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rrtuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Plus,Apply[Times,Union@@mps/@Union[Subsets[q]],{2}],{1}]]];
Table[Length[rrtuks[n]],{n,12}]

a(13)-a(37) from Sean A. Irvine, Jul 13 2022

A320055 Heinz numbers of sum-product knapsack partitions.

Original entry on oeis.org

1, 2, 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

Gus Wiseman, Oct 04 2018

nonn

A 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 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 A320056 in having 2, 845, ... and lacking 245, 455, 847, ....

			A complete list of sums of products of multiset partitions of submultisets of the partition (6,6,3) is:
            0 = 0
          (3) = 3
          (6) = 6
        (3*6) = 18
        (6*6) = 36
      (3*6*6) = 108
      (3)+(6) = 9
    (3)+(6*6) = 39
      (6)+(6) = 12
    (6)+(3*6) = 24
  (3)+(6)+(6) = 15
These are all distinct, and the Heinz number of (6,6,3) is 845, so 845 belongs to the sequence.

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

Cf. A267597, A320052, A320053, A320054, A320056, A320057, A320058.

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,Join@@facs/@Divisors[#]}]&]

A320056 Heinz numbers of product-sum knapsack partitions.

Original entry on oeis.org

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

Gus Wiseman, Oct 04 2018

nonn

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, ....

			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.

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

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

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[#]}]&]

A320052 Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 4, 6, 8, 8

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of product-sum knapsack partitions begins:
   0: ()
   1:
   2: (2)
   3: (3)
   4: (4)
   5: (5) (3,2)
   6: (6) (4,2) (3,3)
   7: (7) (5,2) (4,3)
   8: (8) (6,2) (5,3) (4,4)
   9: (9) (7,2) (6,3) (5,4)
  10: (10) (8,2) (7,3) (6,4) (5,5) (4,3,3)
  11: (11) (9,2) (8,3) (7,4) (6,5) (5,4,2) (5,3,3) (4,4,3)
  12: (12) (10,2) (9,3) (8,4) (7,5) (7,3,2) (6,6) (4,4,4)
A complete list of all products of sums of multiset partitions of submultisets of (4,3,3) is:
           () = 1
          (3) = 3
          (4) = 4
        (3+3) = 6
        (3+4) = 7
      (3+3+4) = 10
      (3)*(3) = 9
      (3)*(4) = 12
    (3)*(3+4) = 21
    (4)*(3+3) = 24
  (3)*(3)*(4) = 36
These are all distinct, so (4,3,3) is a product-sum knapsack partition of 10.

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320053, A320054, A320055, A320056, A320057, A320058.

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rrsuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Times,Apply[Plus,Union@@mps/@Union[Subsets[q]],{2}],{1}]]];
Table[Length[rrsuks[n]],{n,12}]

A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 6, 8, 9, 12, 14

Offset: 0

Gus Wiseman, Oct 04 2018

			The sequence of spanning sum-product knapsack partitions begins:
  0: ()
  1: (1)
  2: (2) (1,1)
  3: (3) (2,1) (1,1,1)
  4: (4) (3,1)
  5: (5) (4,1) (3,2)
  6: (6) (5,1) (4,2) (3,3)
  7: (7) (6,1) (5,2) (4,3) (3,3,1)
  8: (8) (7,1) (6,2) (5,3) (4,4) (3,3,2)
  9: (9) (8,1) (7,2) (6,3) (5,4) (4,4,1) (4,3,2) (3,3,3)
A complete list of all sums of products covering the parts of (3,3,3,2) is:
        (2*3*3*3) = 54
      (2)+(3*3*3) = 29
      (3)+(2*3*3) = 21
      (2*3)+(3*3) = 15
    (2)+(3)+(3*3) = 14
    (3)+(3)+(2*3) = 12
  (2)+(3)+(3)+(3) = 11
These are all distinct, so (3,3,3,2) is a spanning sum-product knapsack partition of 11.
An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.

Cf. A267597, A320052, A320054, A320055, A320056, A320057, A320058.

sps[{}]:={{}};
sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
rtuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Plus,Apply[Times,mps[q],{2}],{1}]]];
Table[Length[rtuks[n]],{n,8}]

A320057 Heinz numbers of spanning sum-product knapsack partitions.

Original entry on oeis.org

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

Gus Wiseman, Oct 04 2018

nonn

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, ...

			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.

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

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

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[#]}]&]

A320058 Heinz numbers of spanning product-sum knapsack partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Oct 04 2018

nonn

A spanning product-sum knapsack partition is a finite multiset m of positive integers such that every product of sums 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 A320057 in having 20, 28, 42, 44, 52, ... and lacking 1155, 1625, 1815, 1875, 1911, ....

			The sequence of all spanning product-sum knapsack partitions begins: (), (1), (2), (1,1), (3), (2,1), (4), (1,1,1), (3,1), (5), (6), (4,1), (3,2), (7), (8), (3,1,1), (4,2), (5,1), (9), (3,3), (6,1), (4,1,1).
A complete list of products of sums of multiset partitions of the partition (3,1,1) is:
      (1+1+3) = 5
    (1)*(1+3) = 4
    (3)*(1+1) = 6
  (1)*(1)*(3) = 3
These are all distinct, and the Heinz number of (3,1,1) is 20, so 20 belongs to the sequence.

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

Cf. A267597, A320052, A320053, A320054, A320055, A320056, A320057.

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,facs[#]}]&]

cp's OEIS Frontend

A267597 Number of sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of any submultiset of y is distinct.

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A320055 Heinz numbers of sum-product knapsack partitions.

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A320056 Heinz numbers of product-sum knapsack partitions.

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A320052 Number of product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums of the parts of a multiset partition of any submultiset of y is distinct.

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A320053 Number of spanning sum-product knapsack partitions of n. Number of integer partitions y of n such that every sum of products of the parts of a multiset partition of y is distinct.

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A320057 Heinz numbers of spanning sum-product knapsack partitions.

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A320058 Heinz numbers of spanning product-sum knapsack partitions.

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