A365921 -id:A365921 - cp's OEIS frontend

A366128 Least non-subset-sum of the multiset of prime indices of n.

0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 3, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 1, 2, 1, 0, 1, 2, 1, 3, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 4, 1, 3, 1, 2, 1, 0, 1, 2, 1, 3, 1, 4, 1, 0, 1, 2, 1, 0, 1, 2, 1

Offset: 1

Gus Wiseman, Oct 06 2023

nonn

Least positive integer up to the sum of prime indices of n that is not the sum of prime indices of any divisor of n, or 0 if none exists.

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 3906 are {1,2,2,4,11}, with least non-subset-sum 10, so a(3906) = 10.

Positions of ones are A005408.
Positions of twos appear to be A091999.
Zeros are A325781, nonzeros A325798.
For greatest instead of least we have A365920 (Frobenius number).
The triangle for this rank statistic is A365921 (partitions with least non-subset-sum k).
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A079068, A257993, A286469, A286470, A339662, A339886.

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[If[nmz[prix[n]]=={},0,Min@@nmz[prix[n]]],{n,100}]

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.

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

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

A366128 Least non-subset-sum of the multiset of prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica