cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A319318 Number of integer partitions of n such that every distinct submultiset has a different GCD.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 2, 6, 5, 5, 5, 8, 5, 9, 6, 8, 9, 11, 6, 11, 11, 11, 10, 14, 9, 16, 12, 14, 15, 15, 11, 19, 17, 17, 14, 22, 15, 22, 18, 18, 21, 25, 16, 24, 21, 23, 22, 28, 21, 26, 22, 26, 27, 32, 20, 35, 30, 27, 27, 31, 27, 38, 30, 33, 29
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2018

Keywords

Comments

Note that such partitions are necessarily strict.

Examples

			The a(31) = 16 partitions are (31), (16,15), (17,14), (18,13), (19,12), (20,11), (21,10), (22,9), (23,8), (24,7), (25,6), (26,5), (27,4), (28,3), (29,2), (15,10,6).

Crossrefs

Cf. A108917, A275972, A289508, A289509, A316313, A319315, A319319, A319320.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@GCD@@@Union[Rest[Subsets[#]]]&]],{n,30}]

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.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Oct 04 2018

Keywords

Examples

			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.

Links

Crossrefs

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.
Cf. A320052, A320053, A320054, A320055, A320056, A320057, A320058.

Programs

  • Mathematica
    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}]

Extensions

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

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

Views

Author

Gus Wiseman, Oct 04 2018

Keywords

Examples

			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.

Crossrefs

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.
Cf. A267597, A320053, A320054, A320055, A320056, A320057, A320058.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Oct 04 2018

Keywords

Examples

			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).

Crossrefs

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.
Cf. A267597, A320052, A320054, A320055, A320056, A320057, A320058.

Programs

  • Mathematica
    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}]

A320054 Number of spanning product-sum knapsack partitions of n. Number of integer partitions y of n such that every product of sums the parts of a multiset partition of y is distinct.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 5, 8, 10, 12, 16, 17, 25
Offset: 0

Views

Author

Gus Wiseman, Oct 04 2018

Keywords

Examples

			The sequence of spanning product-sum 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) (3,1,1)
6: (6) (5,1) (4,2) (4,1,1) (3,3)
7: (7) (6,1) (5,2) (5,1,1) (4,3) (4,2,1) (4,1,1,1) (3,3,1)
8: (8) (7,1) (6,2) (6,1,1) (5,3) (5,2,1) (5,1,1,1) (4,4) (4,3,1) (3,3,2)
9: (9) (8,1) (7,2) (7,1,1) (6,3) (6,2,1) (6,1,1,1) (5,4) (5,3,1) (4,4,1) (4,3,2) (3,3,3)
A complete list of all products of sums covering the parts of (4,1,1,1) is:
        (1+1+1+4) = 7
      (1)*(1+1+4) = 6
      (4)*(1+1+1) = 12
      (1+1)*(1+4) = 10
    (1)*(1)*(1+4) = 5
    (1)*(4)*(1+1) = 8
  (1)*(1)*(1)*(4) = 4
These are all distinct, so (4,1,1,1) is a spanning product-sum knapsack partition of 7.
A complete list of all products of sums covering the parts of (5,3,1,1) is:
        (1+1+3+5) = 10
      (1)*(1+3+5) = 9
      (3)*(1+1+5) = 21
      (5)*(1+1+3) = 25
      (1+1)*(3+5) = 16
      (1+3)*(1+5) = 24
    (1)*(1)*(3+5) = 8
    (1)*(3)*(1+5) = 18
    (1)*(5)*(1+3) = 20
    (3)*(5)*(1+1) = 30
  (1)*(1)*(3)*(5) = 15
These are all distinct, so (5,3,1,1) is a spanning product-sum knapsack partition of 10.
An example of a spanning sum-product knapsack partition that is not a spanning product-sum knapsack partition is (5,4,3,2).

Crossrefs

Cf. A001970, A066739, A108917, A275972, A292886, A316313, A318949, A319318, A319320, A319910, A319913.
Cf. A267597, A320052, A320053, A320055, A320056, A320057, A320058.

Programs

  • Mathematica
    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]]]];
rsuks[n_]:=Select[IntegerPartitions[n],Function[q,UnsameQ@@Apply[Times,Apply[Plus,mps[q],{2}],{1}]]];
Table[Length[rsuks[n]],{n,10}]

A319327 Heinz numbers of integer partitions such that every distinct submultiset has a different LCM.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2018

Keywords

Comments

Note that such a Heinz number is necessarily squarefree, as such a partition is necessarily strict.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
First differs from A304713 (Heinz numbers of pairwise indivisible partitions) at A304713(642) = 2093, which is absent from this sequence because its prime indices are {4,6,9} and LCM(4,9) = LCM(4,6,9) = 36.

Examples

			The sequence of partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (3), (4), (5), (6), (3,2), (7), (8), (9), (10), (11), (5,2), (4,3), (12), (13), (14), (15), (7,2).

Crossrefs

Cf. A056239, A108917, A122768, A275972, A299702, A301899, A301900, A304713, A316313, A319315, A319319, A319320.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@LCM@@@Union[Subsets[primeMS[#]]]&]
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.