A276024 -id:A276024 - cp's OEIS frontend

A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 6, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 9, 4, 4, 4, 6, 2, 8, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6, 2, 8, 2, 6, 6

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

Although the average of an empty set is technically indeterminate, we consider it to be distinct from the other subset-averages.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(60) = 9 distinct subset-averages of (3,2,1,1) are 0/0, 1, 4/3, 3/2, 5/3, 7/4, 2, 5/2, 3.

Cf. A032302, A056239, A108917, A275972, A276024, A296150, A299701, A301899, A301957, A316313.

One more than A316314.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Mean/@Subsets[primeMS[n]]]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A316398(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = A056239(d)/bigomega(d)), mapput(m,s,s); k++)); (1+k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316314(n) + 1.

More terms from Antti Karttunen, Sep 23 2018

A321143 Number of non-isomorphic knapsack multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 10, 31, 87, 272, 835, 2673, 8805, 29583

Offset: 0

Gus Wiseman, Oct 28 2018

A multiset partition is knapsack if every distinct submultiset of the parts has a different multiset union.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 31 knapsack multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}
         {{1},{2}}  {{1},{1,1}}    {{1,2,3,3}}
                    {{1},{2,2}}    {{1,2,3,4}}
                    {{1},{2,3}}    {{1},{1,1,1}}
                    {{2},{1,2}}    {{1,1},{1,1}}
                    {{1},{1},{1}}  {{1},{1,2,2}}
                    {{1},{2},{2}}  {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1,2},{1,2}}
                                   {{1},{2,2,2}}
                                   {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
Missing from this list are {{1},{1},{1,1}} and {{1},{2},{1,2}}, which are not knapsack.

Cf. A002219, A006827, A007716, A108917, A275972, A276024, A292886, A316983, A319616.

A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 10, 10, 24, 24, 43, 42, 88, 72, 136, 122, 242, 213, 392, 320, 630, 490, 916, 742, 1432, 1160, 1955, 1604, 2826, 2310, 3850, 2888, 5416, 4426, 7332, 5814, 10046, 7983, 12946, 10236, 17780, 14100, 22674, 17582, 30232, 23674, 37522, 29426, 49832

Offset: 0

Gus Wiseman, Jun 02 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

The contiguous case is A325676.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,1,1)  (4,1)        (4,2)
                                  (1,1,3)      (5,1)
                                  (1,2,2)      (1,1,4)
                                  (2,2,1)      (2,2,2)
                                  (3,1,1)      (4,1,1)
                                  (1,1,1,1,1)  (1,1,1,1,1,1)

These compositions are ranked by A334967.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702, while the strict case is counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A003022, A029931, A103295, A143823, A325680, A333224.

b:= proc(n, s) option remember; `if`(n=0, 1, add((h->
      `if`(nops(h)=nops(map(l-> add(i, i=l), h)),
       b(n-j, h), 0))({s[], map(l-> [l[], j], s)[]}), j=1..n))
    end:
a:= n-> b(n, {[]}):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 03 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15}]

a(18)-a(47) from Alois P. Heinz, Jun 03 2020

A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 4, 6, 11, 15, 25, 35, 53, 72, 108

Offset: 1

Gus Wiseman, Sep 30 2023

A non-subset-sum of a multiset of positive integers summing to n is an element of {1..n} that is not the sum of any submultiset. A multiset is incomplete if it has at least one non-subset-sum.

			The non-subset-sums of y = {2,2,3} are {1,6}, with maximum 6, so y is counted under a(6).
The a(1) = 1 through a(6) = 15 multisets:
  {2}  {3}    {4}      {5}        {6}          {7}
       {1,3}  {1,4}    {1,5}      {1,6}        {1,7}
              {2,2}    {2,3}      {2,4}        {2,5}
              {1,1,4}  {1,1,5}    {3,3}        {3,4}
                       {1,2,5}    {1,1,6}      {1,1,7}
                       {1,1,1,5}  {1,2,6}      {1,2,7}
                                  {1,3,3}      {1,3,4}
                                  {2,2,2}      {2,2,3}
                                  {1,1,1,6}    {1,1,1,7}
                                  {1,1,2,6}    {1,1,2,7}
                                  {1,1,1,1,6}  {1,1,3,7}
                                               {1,2,2,7}
                                               {1,1,1,1,7}
                                               {1,1,1,2,7}
                                               {1,1,1,1,1,7}

For least instead of greatest we have A126796, ranks A325781, strict A188431.
These multisets have ranks A365830.
Counts appearances of n in the rank statistic A365920.
Column sums of A365921.
These multisets counted by sum are A365924, strict A365831.
The strict case is A366129.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365918 counts non-subset-sums of partitions.
A365923 counts partitions by non-subset sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365658, A365919, A365925.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n],Max@@nmz[#]==n&]],{n,5}]

A366737 Number of numbers k <= A056239(n) that can be written as a linear combination of the prime indices of n (allowing coefficients of 0).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 5, 4, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 6, 1, 6, 1, 5, 5, 8, 4, 6, 1, 9, 4, 6, 1, 7, 1, 7, 6, 10, 1, 6, 2, 7, 6, 8, 1, 7, 4, 7, 5, 11, 1, 7, 1, 12, 4, 6, 3, 8, 1, 9, 7, 8, 1, 7, 1, 13, 7, 10, 4, 9, 1, 7, 4, 14, 1, 8, 5

Offset: 1

Gus Wiseman, Oct 19 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 33 are {2,5}, with combinations
  2 = 2
  4 = 2+2
  5 = 5
  6 = 2+2+2
  7 = 5+2
Hence a(33) = 5.

For minimum instead of length we have A055396.
Positions of first appearances are 1, 2, and A100484.
For subsets instead of combinations we have A304793, complement A325799.
A056239 adds up prime indices, row sums of A112798.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A018818, A046663, A055932, A299701, A365658, A365918, A365921.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Range[Total[prix[n]]],combs[#,prix[n]]!={}&]],{n,30}]

a(2n) = A056239(2n) - 1 for n > 0.

A305613 Numbers whose multiset of prime factors is not knapsack.

Original entry on oeis.org

30, 60, 70, 72, 84, 90, 120, 140, 144, 150, 168, 180, 210, 216, 240, 252, 270, 280, 286, 288, 300, 308, 330, 336, 350, 360, 378, 390, 420, 432, 440, 450, 480, 490, 495, 504, 510, 525, 528, 540, 560, 570, 572, 576, 588, 594, 600, 616, 630, 646, 648, 660, 672

Offset: 1

Gus Wiseman, Jun 06 2018

nonn

A multiset of positive integers is knapsack if every distinct submultiset has a different sum.

			30 = 2 * 3 * 5 is not knapsack because 2 + 3 = 5.

Cf. A027746, A108917, A122768, A275972, A276024, A299701, A299729, A301855, A301900, A304793, A305611.

Select[Range[1000],DivisorSigma[0,#]=!=Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[#],{p_,k_}:>Table[p,{k}]]]]]&]

A316362 Heinz numbers of strict integer partitions such that not every distinct subset has a different average.

Original entry on oeis.org

30, 105, 110, 210, 238, 273, 330, 385, 390, 462, 506, 510, 546, 570, 627, 690, 714, 770, 806, 858, 870, 910, 930, 935, 966, 1001, 1110, 1131, 1155, 1190, 1230, 1254, 1290, 1326, 1330, 1365, 1394, 1410, 1430, 1482, 1495, 1518, 1590, 1729, 1770, 1785, 1786, 1794

Offset: 1

Gus Wiseman, Jun 30 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			462 is the Heinz number of (5,4,2,1), and the subsets {1,5}, and {2,4} have the same average, so 462 belongs to the sequence.

Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299702, A301899, A316313, A316314, A316361.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[3000],SquareFreeQ[#]&&!UnsameQ@@Mean/@Union[Subsets[primeMS[#]]]&]

A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 7, 11, 11, 15, 18, 23, 28, 36, 40, 50, 59, 70, 83, 101, 118, 141, 166, 195, 227, 268, 306, 358, 414, 478, 549, 640, 730, 846, 968, 1113, 1271, 1462, 1657, 1897, 2154, 2451

Offset: 1

Gus Wiseman, Oct 07 2023

A non-subset-sum of a set summing to n is a positive integer up to n that is not the sum of any subset. For example, the non-subset-sums of {1,3,4} are {2,6}.

			The a(1) = 1 through a(8) = 11 sets:
  {2}  {3}    {4}    {5}      {6}      {7}      {8}        {9}
       {1,3}  {1,4}  {2,3}    {2,4}    {2,5}    {2,6}      {2,7}
                     {1,5}    {1,6}    {3,4}    {3,5}      {3,6}
                     {1,2,5}  {1,2,6}  {1,7}    {1,8}      {4,5}
                                       {1,3,4}  {1,3,5}    {2,3,4}
                                       {1,2,7}  {1,2,8}    {1,9}
                                                {1,2,3,8}  {1,3,6}
                                                           {1,4,5}
                                                           {1,2,9}
                                                           {1,2,3,9}
                                                           {1,2,4,9}

For least instead of greatest: A188431, non-strict A126796 (ranks A325781).
The version counting multisets instead of sets is A366127.
These sets counted by sum are A365924, strict A365831.
A046663 counts partitions without a submultiset summing k, strict A365663.
A325799 counts non-subset-sums of prime indices.
A365923 counts partitions by number of non-subset-sums, strict A365545.
Cf. A006827, A276024, A284640, A304792, A365543, A365658, A365661, A365918, A365920, A365921, A365925.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n,2*n], UnsameQ@@#&&Max@@nmz[#]==n&]],{n,15}]

a(31)-a(42) from Erich Friedman, Nov 13 2024

A326033 Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 1, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 4, 5, 0, 30, 2, 9, 9, 20, 3, 37, 6, 18, 16, 37, 20, 71, 12, 37, 40

Offset: 1

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

			The partition (10,8,6,6) is counted under a(30) because (10,10,8,6,6), (10,8,8,6,6), and (10,8,6,6,6) are not knapsack.

Cf. A002033, A108917, A275972, A276024, A299702.

Cf. A325857, A325862, A325863, A325864, A325865, A326015, A326016, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Union[#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]

A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 0, 3, 3, 2, 1, 1, 0, 0, 0, 0, 4, 5, 3, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 3, 2, 1, 1, 0, 0, 0, 0, 4, 8, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 2, 9, 9, 7, 5

Offset: 0

Gus Wiseman, Nov 09 2023

An integer partition of n is complete (ranks A325781) if every integer from 0 to n is the sum of some submultiset of the parts.

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  0  1  1
  0  0  0  2  1  1
  0  0  0  1  2  1  1
  0  0  0  1  3  2  1  1
  0  0  0  0  3  3  2  1  1
  0  0  0  0  4  5  3  2  1  1
  0  0  0  0  3  5  5  3  2  1  1
  0  0  0  0  4  8  7  5  3  2  1  1
  0  0  0  0  2  9  9  7  5  3  2  1  1
  0  0  0  0  2 11 12 11  7  5  3  2  1  1
  0  0  0  0  1 11 16 13 11  7  5  3  2  1  1
  0  0  0  0  1 14 21 19 15 11  7  5  3  2  1  1
Row n = 11 counts the following partitions (empty columns not shown):
  6311  62111  611111  5111111  41111111  311111111  2111111111  11111111111
  6221  53111  521111  4211111  32111111  221111111
  5321  52211  431111  3311111  22211111
  4421  44111  422111  3221111
        43211  332111  2222111
        42221  322211
        33311  222221
        33221

Column k appears to have A000325(k) nonzero terms.
Column sums are A003513.
Central column T(2n,n) is A007042.
Row sums are A126796, ranks A325781.
The strict case is too sparse, row sums A188431 (complement A365831).
Grouping by maximum instead of length gives A261036.
A000041 counts integer partitions.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts subset-sums of prime indices, firsts A259941.
A365924 counts incomplete partitions, ranks A365830.
Cf. A002033, A018818, A046663, A047967, A055932, A276024, A304792, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n,{k}],nmz[#]=={}&]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A316398 Number of distinct subset-averages of the integer partition with Heinz number n.

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A321143 Number of non-isomorphic knapsack multiset partitions of weight n.

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A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

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A366127 Number of finite incomplete multisets of positive integers with greatest non-subset-sum n.

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A366737 Number of numbers k <= A056239(n) that can be written as a linear combination of the prime indices of n (allowing coefficients of 0).

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A305613 Numbers whose multiset of prime factors is not knapsack.

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A316362 Heinz numbers of strict integer partitions such that not every distinct subset has a different average.

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A366129 Number of finite sets of positive integers with greatest non-subset-sum n.

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A326033 Number of knapsack partitions of n such that no addition of one part equal to an existing part is knapsack.