A325988 -id:A325988 - cp's OEIS frontend

A326020 Number of complete subsets of {1..n}.

1, 2, 3, 4, 6, 9, 15, 27, 50, 95, 185, 365, 724, 1441, 2873, 5735, 11458, 22902, 45789, 91561, 183102, 366180, 732331, 1464626, 2929209, 5858367, 11716674, 23433277, 46866473, 93732852, 187465596, 374931067, 749861989, 1499723808, 2999447418

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,3,4}
                                      {1,2,3,5}    {1,2,3,5}
                                      {1,2,4,5}    {1,2,3,6}
                                      {1,2,3,4,5}  {1,2,4,5}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,4,5,6}
                                                   {1,2,3,4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..300
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326021, A326022, A326036.

Table[Length[Select[Subsets[Range[n]],Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,0,10}]

a(17)-a(34) from Charlie Neder, Jun 05 2019

A325986 Heinz numbers of complete strict integer partitions.

Original entry on oeis.org

1, 2, 6, 30, 42, 210, 330, 390, 462, 510, 546, 714, 798, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7854, 8778, 8970, 9282, 9570, 9690, 10230, 10374, 10626, 11310, 11730, 12090, 12210, 12558, 13398, 13566, 14322, 14430

Offset: 1

Gus Wiseman, May 30 2019

nonn

Strict partitions are counted by A000009, while complete partitions are counted by A126796.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition of n is complete (A126796, A325781) if every number from 0 to n is the sum of some submultiset of the parts.
The enumeration of these partitions by sum is given by A188431.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     42: {1,2,4}
    210: {1,2,3,4}
    330: {1,2,3,5}
    390: {1,2,3,6}
    462: {1,2,4,5}
    510: {1,2,3,7}
    546: {1,2,4,6}
    714: {1,2,4,7}
    798: {1,2,4,8}
   2310: {1,2,3,4,5}
   2730: {1,2,3,4,6}
   3570: {1,2,3,4,7}
   3990: {1,2,3,4,8}
   4290: {1,2,3,5,6}
   4830: {1,2,3,4,9}
   5610: {1,2,3,5,7}

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A304793.

Cf. A325780, A325781, A325782, A325788, A325790, A325988.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Select[Range[1000],SquareFreeQ[#]&&Union[hwt/@Divisors[#]]==Range[0,hwt[#]]&]

Intersection of A005117 (strict partitions) and A325781 (complete partitions).

A326036 Number of uniform complete integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 2, 2, 2, 2, 6, 3, 3, 5, 5, 3, 8, 5, 11, 10, 10, 9, 19, 13, 15, 17, 21, 18, 35, 26, 39, 40, 50, 50, 77, 63, 84, 88, 113, 103, 146, 132, 171, 180, 212, 214, 292, 276, 345, 363, 435, 442, 561, 569, 694, 729, 853, 891, 1108

Offset: 0

Gus Wiseman, Jun 04 2019

nonn

An integer partition of n is uniform if all parts appear with the same multiplicity, and complete if every nonnegative integer up to n is the sum of some submultiset.

			The initial terms count the following partitions:
   0: ()
   1: (1)
   2: (11)
   3: (21)
   3: (111)
   4: (1111)
   5: (11111)
   6: (321)
   6: (2211)
   6: (111111)
   7: (421)
   7: (1111111)
   8: (3311)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (4321)
  10: (1111111111)
  11: (5321)
  11: (11111111111)

Cf. A002033, A047966, A072774, A108917, A126796, A188431, A276024.

Cf. A325781, A325791, A325988, A326020, A326035, A326037.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#]&&Sort[sums[Sort[#]]]==Range[Total[#]]&]],{n,0,30}]

A325990 Numbers with more than one perfect factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 256, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456

Offset: 1

Gus Wiseman, May 30 2019

nonn

First differs from A060476 in lacking 1 and having 432.

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

Positions of terms > 1 in A325989.

Cf. A002033, A292886, A325780, A325781, A325782, A325787, A325789, A325988.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Function[n,Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]]>1]]

A325989 Number of perfect factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 30 2019

nonn

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

			The a(216) = 4 perfect factorizations:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*3*3*3*4)
  (2*3*4*9)

Positions of terms > 1 are A325990.

Cf. A002033, A103300, A292886, A325685, A325780, A325787, A325789, A325988.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]],{n,100}]

a(2^n) = A002033(n).

A326021 Number of complete subsets of {1..n} with maximum n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 23, 45, 90, 180, 359, 717, 1432, 2862, 5723, 11444, 22887, 45772, 91541, 183078, 366151, 732295, 1464583, 2929158, 5858307, 11716603, 23433196, 46866379, 93732744, 187465471, 374930922, 749861819, 1499723610

Offset: 1

Gus Wiseman, Jun 04 2019

nonn

A set of positive integers summing to n is complete if every nonnegative integer up to n is the sum of some subset.

			The a(1) = 1 through a(7) = 12 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,4}    {1,2,3,5}    {1,2,3,6}      {1,2,3,7}
                       {1,2,3,4}  {1,2,4,5}    {1,2,4,6}      {1,2,4,7}
                                  {1,2,3,4,5}  {1,2,3,4,6}    {1,2,3,4,7}
                                               {1,2,3,5,6}    {1,2,3,5,7}
                                               {1,2,4,5,6}    {1,2,3,6,7}
                                               {1,2,3,4,5,6}  {1,2,4,5,7}
                                                              {1,2,4,6,7}
                                                              {1,2,3,4,5,7}
                                                              {1,2,3,4,6,7}
                                                              {1,2,3,5,6,7}
                                                              {1,2,4,5,6,7}
                                                              {1,2,3,4,5,6,7}

Charlie Neder, Table of n, a(n) for n = 1..300
Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326022, A326036.

Table[Length[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]],{n,10}]

a(18)-a(34) from Charlie Neder, Jun 05 2019

A326022 Number of minimal complete subsets of {1..n} with maximum n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 4, 8, 8, 8, 10, 14, 25, 40, 49, 62

Offset: 1

Gus Wiseman, Jun 04 2019

A set of positive integers summing to m is complete if every nonnegative integer up to m is the sum of some subset. For example, (1,2,3,6,13) is a complete set because we have:
= (empty sum)
= 1
= 2
= 3
= 1 + 3
= 2 + 3
= 6
= 6 + 1
= 6 + 2
= 6 + 3
= 1 + 3 + 6
= 2 + 3 + 6
= 1 + 2 + 3 + 6
and the remaining numbers 13-25 are obtained by adding 13 to each of these.

			The a(3) = 1 through a(9) = 8 subsets:
  {1,2,3}  {1,2,4}  {1,2,3,5}  {1,2,3,6}  {1,2,3,7}  {1,2,4,8}    {1,2,3,4,9}
                    {1,2,4,5}  {1,2,4,6}  {1,2,4,7}  {1,2,3,5,8}  {1,2,3,5,9}
                                                     {1,2,3,6,8}  {1,2,3,6,9}
                                                     {1,2,3,7,8}  {1,2,3,7,9}
                                                                  {1,2,4,5,9}
                                                                  {1,2,4,6,9}
                                                                  {1,2,4,7,9}
                                                                  {1,2,4,8,9}

Andrzej Kukla and Piotr Miska, On practical sets and A-practical numbers, arXiv:2405.18225 [math.NT], 2024.

Cf. A002033, A103295, A108917, A126796, A188431, A276024.

Cf. A325684, A325781, A325790, A325791, A325986, A325988, A326016, A326020, A326021, A326036.

fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]&/@Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Select[Subsets[Range[n]],Max@@#==n&&Union[Plus@@@Subsets[#]]==Range[0,Total[#]]&]]],{n,10}]

cp's OEIS Frontend

A326020 Number of complete subsets of {1..n}.

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A325986 Heinz numbers of complete strict integer partitions.

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A326036 Number of uniform complete integer partitions of n.

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A325990 Numbers with more than one perfect factorization.

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A325989 Number of perfect factorizations of n.

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A326021 Number of complete subsets of {1..n} with maximum n.

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A326022 Number of minimal complete subsets of {1..n} with maximum n.

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