A365046 -id:A365046 - cp's OEIS frontend

A364914 Number of subsets of {1..n} such that some element can be written as a nonnegative linear combination of the others.

0, 0, 1, 3, 9, 20, 48, 101, 219, 454, 944, 1917, 3925, 7915, 16004, 32188, 64751, 129822, 260489, 521672, 1045060, 2091808, 4187047, 8377255, 16762285, 33531228, 67077485, 134170217, 268371678, 536772231, 1073611321, 2147282291, 4294697258, 8589527163, 17179321094

Offset: 0

Gus Wiseman, Aug 17 2023

nonn

A variation of non-binary combination-full sets where parts can be re-used. The complement is counted by A326083. The binary version is A093971. For non-re-usable parts we have A364534. First differences are A365046.

			The set {3,4,5,17} has 17 = 1*3 + 1*4 + 2*5, so is counted under a(17).
The a(0) = 0 through a(5) = 20 subsets:
  .  .  {1,2}  {1,2}    {1,2}      {1,2}
               {1,3}    {1,3}      {1,3}
               {1,2,3}  {1,4}      {1,4}
                        {2,4}      {1,5}
                        {1,2,3}    {2,4}
                        {1,2,4}    {1,2,3}
                        {1,3,4}    {1,2,4}
                        {2,3,4}    {1,2,5}
                        {1,2,3,4}  {1,3,4}
                                   {1,3,5}
                                   {1,4,5}
                                   {2,3,4}
                                   {2,3,5}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The binary complement is A007865.
The binary version without re-usable parts is A088809.
The binary version is A093971.
The complement without re-usable parts is A151897.
The complement is counted by A326083.
The version without re-usable parts is A364534.
The version for strict partitions is A364839, complement A364350.
The version for partitions is A364913.
The version for positive combinations is A365043, complement A365044.
First differences are A365046.
Cf. A011782, A085489, A103580, A116861, A124506, A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364756.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A364914(n):
    c, mlist = 0, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m,k=m-1):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(2,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for d in w:
                for s in mlist[d-1]:
                    if s <= ws:
                        c += 1
                        break
                else:
                    continue
                break
    return c # Chai Wah Wu, Nov 17 2023

a(12)-a(34) from Chai Wah Wu, Nov 17 2023

A124506 Number of numerical semigroups with Frobenius number n; that is, numerical semigroups for which the largest integer not belonging to them is n.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 11, 10, 21, 22, 51, 40, 106, 103, 200, 205, 465, 405, 961, 900, 1828, 1913, 4096, 3578, 8273, 8175, 16132, 16267, 34903, 31822, 70854, 68681, 137391, 140661, 292081, 270258, 591443, 582453, 1156012

Offset: 1

P. A. Garcia-Sanchez (pedro(AT)ugr.es), Dec 18 2006

From Gus Wiseman, Aug 28 2023: (Start)
Appears to be the number of subsets of {1..n} containing n such that no element can be written as a nonnegative linear combination of the others, first differences of A326083. For example, the a(1) = 1 through a(8) = 10 subsets are:
  {1}  {2}  {3}    {4}    {5}      {6}      {7}        {8}
            {2,3}  {3,4}  {2,5}    {4,6}    {2,7}      {3,8}
                          {3,5}    {5,6}    {3,7}      {5,8}
                          {4,5}    {4,5,6}  {4,7}      {6,8}
                          {3,4,5}           {5,7}      {7,8}
                                            {6,7}      {3,7,8}
                                            {3,5,7}    {5,6,8}
                                            {4,5,7}    {5,7,8}
                                            {4,6,7}    {6,7,8}
                                            {5,6,7}    {5,6,7,8}
                                            {4,5,6,7}
Note that these subsets do not all generate numerical semigroups, as their GCD is unrestricted, cf. A358392. The complement is counted by A365046, first differences of A364914.
(End)

			a(1) = 1 via <2,3> = {0,2,3,4,...}; the largest missing number is 1.
a(2) = 1 via <3,4,5> = {0,3,4,5,...}; the largest missing number is 2.
a(3) = 2 via <2,5> = {0,2,4,5,...}; and <4,5,6,7> = {0,4,5,6,7,...} where in both the largest missing number is 3.
a(4) = 2 via <3,5,7> = {0,3,5,6,7,...} and <5,6,7,8,9> = {5,6,7,8,9,...} where in both the largest missing number is 4.

S. R. Finch, Monoids of natural numbers
Manuel Delgado, Neeraj Kumar, and Claude Marion, On counting numerical semigroups by maximum primitive and Wilf's conjecture, arXiv:2501.04417 [math.CO], 2025. See p. 22.
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia, and J. A. Jimenez-Madrid, Fundamental gaps in numerical semigroups, Journal of Pure and Applied Algebra 189 (2004) 301-313.
Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).

Cf. A158206. [From Steven Finch, Mar 13 2009]
A288728 counts sum-free sets, first differences of A007865.
A364350 counts combination-free partitions, complement A364839.
Cf. A085489, A088809, A093971, A103580, A116861, A151897, A237668, A308546, A326020, A326083, A364349, A365069.

The sequence was originally generated by a C program and a Haskell script. The sequence can be obtained by using the function NumericalSemigroupsWithFrobeniusNumber included in the numericalsgps GAP package.

A088314 Cardinality of set of sets of parts of all partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 51, 61, 79, 96, 124, 148, 186, 222, 275, 326, 400, 473, 575, 673, 811, 946, 1132, 1317, 1558, 1813, 2138, 2463, 2893, 3323, 3882, 4461, 5177, 5917, 6847, 7818, 8994, 10251, 11766, 13334, 15281, 17309, 19732, 22307

Offset: 0

Naohiro Nomoto, Nov 05 2003

Number of different values of A007947(m) when A056239(m) is equal to n.
From Gus Wiseman, Sep 11 2023: (Start)
Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:
  {1}  {1}  {1}    {1}    {1}    {1}      {1}
       {2}  {3}    {2}    {5}    {2}      {7}
            {1,2}  {4}    {1,2}  {3}      {1,2}
                   {1,2}  {1,3}  {6}      {1,3}
                   {1,3}  {1,4}  {1,2}    {1,4}
                          {2,3}  {1,3}    {1,5}
                                 {1,4}    {1,6}
                                 {1,5}    {2,3}
                                 {2,4}    {2,5}
                                 {1,2,3}  {3,4}
                                          {1,2,3}
                                          {1,2,4}
Cf. A365073, A365311, A365312, A365322, A365380.
(End)

			The 7 partitions of 5 and their sets of parts are
[ #]  partition      set of parts
[ 1]  [ 1 1 1 1 1 ]  {1}
[ 2]  [ 2 1 1 1 ]    {1, 2}
[ 3]  [ 2 2 1 ]      {1, 2}  (same as before)
[ 4]  [ 3 1 1 ]      {1, 3}
[ 5]  [ 3 2 ]        {2, 3}
[ 6]  [ 4 1 ]        {1, 4}
[ 7]  [ 5 ]          {5}
so we have a(5) = |{{1}, {1, 2}, {1, 3}, {2, 3}, {1, 4}, {5}}| = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A182410.
The complement in subsets of {1..n-1} is A070880(n) = A365045(n) - 1.
The case of pairs is A365315, see also A365314, A365320, A365321.
A116861 and A364916 count linear combinations of strict partitions.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364914 and A365046 count combination-full subsets.
Cf. A000009, A007865, A088528, A088809, A093971, A326020, A364272, A364534, A365043, A364350.

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

list2set := L -> {op(L)};
a:= N -> list2set(map( list2set, combinat[partition](N) ));
seq(nops(a(n)), n=0..30);
#  Yogy Namara (yogy.namara(AT)gmail.com), Jan 13 2010
b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x->{x[],i}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..40);
# Alois P. Heinz, Aug 09 2012

Table[Length[Union[Map[Union,IntegerPartitions[n]]]],{n,1,30}] (* Geoffrey Critzer, Feb 19 2013 *)
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i < 1, {},
     Union@Flatten@{b[n, i - 1], Table[If[Head[#] == List,
     Append[#, i]]& /@ b[n - i*j, i - 1], {j, 1, n/i}]}]];
a[n_] := Length[b[n, n]];
a /@ Range[0, 40] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
combp[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,1,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&&combp[n,#]!={}&]], {n,0,15}] (* Gus Wiseman, Sep 11 2023 *)

from sympy.utilities.iterables import partitions
def A088314(n): return len({tuple(sorted(set(p))) for p in partitions(n)}) # Chai Wah Wu, Sep 10 2023

a(n) = 2^(n-1) - A070880(n). - Alois P. Heinz, Feb 08 2019

a(n) = A365042(n) + 1. - Gus Wiseman, Sep 13 2023

More terms and clearer definition from Vladeta Jovovic, Apr 21 2005

A365381 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 1, 8, 4, 4, 5, 2, 2, 1, 16, 8, 8, 10, 10, 7, 5, 5, 2, 2, 1, 32, 16, 16, 20, 20, 23, 15, 15, 12, 12, 8, 5, 5, 2, 2, 1, 64, 32, 32, 40, 40, 46, 47, 38, 33, 35, 29, 28, 21, 17, 14, 13, 8, 5, 5, 2, 2, 1, 128, 64, 64, 80, 80, 92, 94, 102, 79, 82, 76, 75, 68, 64, 53, 48, 43, 34, 33, 23, 19, 15, 13, 8, 5, 5, 2, 2, 1

Offset: 0

Gus Wiseman, Sep 08 2023

Row lengths are A000124(n) = 1 + n*(n+1)/2.

			Triangle begins:
   1
   2  1
   4  2  2  1
   8  4  4  5  2  2  1
  16  8  8 10 10  7  5  5  2  2  1
  32 16 16 20 20 23 15 15 12 12  8  5  5  2  2  1
  64 32 32 40 40 46 47 38 33 35 29 28 21 17 14 13  8  5  5  2  2  1
Array begins:
     k=0   k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9
-------------------------------------------------------
n=0:  1
n=1:  2     1
n=2:  4     2    2    1
n=3:  8     4    4    5    2    2    1
n=4:  16    8    8    10   10   7    5    5    2    2
n=5:  32    16   16   20   20   23   15   15   12   12
n=6:  64    32   32   40   40   46   47   38   33   35
n=7:  128   64   64   80   80   92   94   102  79   82
n=8:  256   128  128  160  160  184  188  204  207  184
n=9:  512   256  256  320  320  368  376  408  414  440
The T(5,8) = 12 subsets are:
  {3,5}  {1,2,5}  {1,2,3,4}  {1,2,3,4,5}
         {1,3,4}  {1,2,3,5}
         {1,3,5}  {1,2,4,5}
         {2,3,5}  {1,3,4,5}
         {3,4,5}  {2,3,4,5}

Row lengths are A000124 = number of distinct sums of subsets of {1..n}.
Central column/main diagonal is A365376.
A000009 counts sets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365046 counts combination-full subsets, differences of A364914.
Cf. A007865, A085489, A093971, A103580, A131577, A151897, A326080, A364272, A364534, A365073, A365377, A365380.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],k]&]],{n,0,8},{k,0,n*(n+1)/2}]

A365073 Number of subsets of {1..n} that can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 60, 112, 244, 480, 992, 1944, 4048, 7936, 16176, 32320, 65088, 129504, 261248, 520448, 1046208, 2090240, 4186624, 8365696, 16766464, 33503744, 67064064, 134113280, 268347392, 536546816, 1073575936, 2146703360, 4294425600, 8588476416, 17178349568

Offset: 0

Gus Wiseman, Sep 01 2023

nonn

			The subset {2,3,6} has 7 = 2*2 + 1*3 + 0*6 so is counted under a(7).
The a(1) = 1 through a(4) = 14 subsets:
  {1}  {1}    {1}      {1}
       {2}    {3}      {2}
       {1,2}  {1,2}    {4}
              {1,3}    {1,2}
              {2,3}    {1,3}
              {1,2,3}  {1,4}
                       {2,3}
                       {2,4}
                       {3,4}
                       {1,2,3}
                       {1,2,4}
                       {1,3,4}
                       {2,3,4}
                       {1,2,3,4}

Andrew Howroyd, Table of n, a(n) for n = 0..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The case of positive coefficients is A088314.
The case of subsets containing n is A131577.
The binary version is A365314, positive A365315.
The binary complement is A365320, positive A365321.
The positive complement is counted by A365322.
A version for partitions is A365379, strict A365311.
The complement is counted by A365380.
The case of subsets without n is A365542.
A326083 and A124506 appear to count combination-free subsets.
A179822 and A326080 count sum-closed subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A007865, A088809, A093971, A151897, A237668, A308546, A326020, A364534, A364839, A365043, A365381.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],combs[n,#]!={}&]],{n,0,5}]

a(n)={
  my(comb(k,b)=while(b>>k, b=bitor(b, b>>k); k*=2); b);
  my(recurse(k,b)=
    if(bittest(b,0), 2^(n+1-k),
    if(2*k>n, 2^(n+1-k) - 2^sum(j=k, n, !bittest(b,j)),
    self()(k+1, b) + self()(k+1, comb(k,b)) )));
  recurse(1, 1<Andrew Howroyd, Sep 04 2023

Terms a(12) and beyond from Andrew Howroyd, Sep 04 2023

A365380 Number of subsets of {1..n} that cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 16, 12, 32, 32, 104, 48, 256, 208, 448, 448, 1568, 896, 3840, 2368, 6912, 7680, 22912, 10752, 50688, 44800, 104448, 88064, 324096, 165888, 780288, 541696, 1458176, 1519616, 4044800, 2220032, 10838016, 8744960, 20250624, 16433152, 62267392, 34865152

Offset: 1

Gus Wiseman, Sep 04 2023

nonn

			The set {4,5,6} cannot be linearly combined to obtain 7 so is counted under a(7), but we have 8 = 2*4 + 0*5 + 0*6, so it is not counted under a(8).
The a(1) = 1 through a(8) = 12 subsets:
  {}  {}  {}   {}   {}     {}     {}       {}
          {2}  {3}  {2}    {4}    {2}      {3}
                    {3}    {5}    {3}      {5}
                    {4}    {4,5}  {4}      {6}
                    {2,4}         {5}      {7}
                    {3,4}         {6}      {3,6}
                                  {2,4}    {3,7}
                                  {2,6}    {5,6}
                                  {3,5}    {5,7}
                                  {3,6}    {6,7}
                                  {4,5}    {3,6,7}
                                  {4,6}    {5,6,7}
                                  {5,6}
                                  {2,4,6}
                                  {3,5,6}
                                  {4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 1..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The complement is counted by A365073, without n A365542.
The binary complement is A365314, positive A365315.
The binary case is A365320, positive A365321.
For positive coefficients we have A365322, complement A088314.
A124506 appears to count combination-free subsets, differences of A326083.
A179822 counts sum-closed subsets, first differences of A326080.
A288728 counts binary sum-free subsets, first differences of A007865.
A365046 counts combination-full subsets, first differences of A364914.
A365071 counts sum-free subsets, first differences of A151897.
Cf. A050291, A085489, A088528, A088809, A093971, A326020, A364350, A364534, A365043, A365045.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n-1]],combs[n,#]=={}&]],{n,5}]

a(n) = 2^n - A365073(n).

Terms a(12) and beyond from Andrew Howroyd, Sep 04 2023

A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.

Original entry on oeis.org

1, 2, 3, 5, 10, 20, 40, 82, 169, 348, 716, 1471, 3016, 6171, 12605, 25710, 52370, 106539, 216470, 439310, 890550, 1803415, 3648557, 7375141, 14896184, 30065129, 60639954, 122231740, 246239551, 495790161, 997747182, 2006969629, 4035274292, 8110185100, 16293958314, 32724456982

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

			The a(0) = 1 through a(4) = 10 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {1,2}  {1,2}    {1,2}
                  {2,3}    {2,3}
                  {1,2,3}  {2,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216*   A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A237668 counts sum-full partitions, ranks A364532.
A240855 counts strict partitions whose length is a part, complement A240861.
A364272 counts sum-full strict partitions, sum-free A364349.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts sets containing two distinct elements summing to k.
Cf. A068911, A095944, A103580, A288728, A326080, A326083, A365376, A365544.

Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]

a(n) = 2^n - A367217(n). - Chai Wah Wu, Nov 14 2023

a(16)-a(28) from Chai Wah Wu, Nov 14 2023

a(29)-a(35) from Max Alekseyev, Feb 25 2025

A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.

Original entry on oeis.org

0, 0, 1, 3, 6, 12, 24, 46, 87, 164, 308, 577, 1080, 2021, 3779, 7058, 13166, 24533, 45674, 84978, 158026, 293737, 545747, 1013467, 1881032, 3489303, 6468910, 11985988, 22195905, 41080751, 75994642, 140514019, 259693004, 479749492, 885910870, 1635281386

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

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

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217*   A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A229816 counts partitions whose length is not a part, complement A002865.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A237667 counts sum-free partitions, ranks A364531.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts sets containing two distinct elements summing to k.
Cf. A068911, A103580, A240861, A288728, A326080, A326083, A364346, A364349, A365046, A365376, A365377.

Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,15}]

a(n) = 2^n - A367216(n). - Chai Wah Wu, Nov 14 2023

a(16)-a(28) from Chai Wah Wu, Nov 14 2023

a(29)-a(35) from Max Alekseyev, Feb 25 2025

A367218 Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 8, 13, 18, 26, 35, 50, 66, 92, 119, 160, 208, 275, 350, 457, 579, 742, 933, 1185, 1476, 1859, 2300, 2868, 3531, 4371, 5343, 6575, 8003, 9776, 11842, 14394, 17351, 20987, 25191, 30315, 36257, 43448, 51753, 61776, 73342, 87192, 103184, 122253, 144211

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The Heinz numbers of these partitions are given by A367226.

			The partition (4,2,1) has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(7).
The a(1) = 1 through a(7) = 13 partitions:
  (1)  (11)  (21)   (22)    (32)     (42)      (52)
             (111)  (31)    (41)     (51)      (61)
                    (211)   (221)    (321)     (322)
                    (1111)  (311)    (411)     (331)
                            (2111)   (2211)    (421)
                            (11111)  (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218*   A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A008284 counts partitions by length, strict A008289.
A240855 counts strict partitions whose length is a part, complement A240861.
A365046 counts combination-full subsets, differences of A364914.
Cf. A088314, A363225, A364350, A365073, A365311, A365918.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], combs[Length[#], Union[#]]!={}&]], {n,0,15}]

a(31)-a(48) from Chai Wah Wu, Nov 15 2023

A367219 Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 2, 4, 4, 7, 6, 11, 9, 16, 16, 23, 22, 35, 33, 48, 50, 69, 70, 99, 99, 136, 142, 187, 194, 261, 267, 346, 367, 468, 489, 626, 650, 824, 870, 1081, 1135, 1421, 1485, 1833, 1942, 2374, 2501, 3062, 3220, 3915, 4145, 4987, 5274, 6363, 6709, 8027

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			3 cannot be written as a nonnegative linear combination of 2 and 5, so (5,2,2) is counted under a(9).
The a(2) = 1 through a(10) = 7 partitions:
  (2)  (3)  (4)  (5)  (6)      (7)    (8)      (9)      (10)
                      (3,3)    (4,3)  (4,4)    (5,4)    (5,5)
                      (2,2,2)         (5,3)    (6,3)    (6,4)
                                      (4,2,2)  (5,2,2)  (7,3)
                                                        (4,4,2)
                                                        (6,2,2)
                                                        (2,2,2,2,2)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219*
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A008284 counts partitions by length, strict A008289.
A124506 appears to count combination-free subsets, differences of A326083.
A365046 counts combination-full subsets, differences of A364914.
Cf. A068911, A088314, A116861, A364345, A364350, A365073, A365312, A365380.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],combs[Length[#],Union[#]]=={}&]],{n,0,15}]

a(31)-a(56) from Chai Wah Wu, Nov 15 2023

cp's OEIS Frontend

A364914 Number of subsets of {1..n} such that some element can be written as a nonnegative linear combination of the others.

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A124506 Number of numerical semigroups with Frobenius number n; that is, numerical semigroups for which the largest integer not belonging to them is n.

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A088314 Cardinality of set of sets of parts of all partitions of n.

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A365381 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.

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A365073 Number of subsets of {1..n} that can be linearly combined using nonnegative coefficients to obtain n.

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A365380 Number of subsets of {1..n} that cannot be linearly combined using nonnegative coefficients to obtain n.

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A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.

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A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.