A326080 -id:A326080 - cp's OEIS frontend

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

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}]

A179009 Number of maximally refined partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 3, 5, 1, 3, 2, 3, 5, 7, 2, 5, 3, 4, 6, 7, 11, 3, 8, 5, 6, 6, 8, 11, 15, 7, 13, 9, 9, 9, 10, 12, 16, 22, 11, 20, 15, 17, 14, 15, 16, 18, 24, 30, 18, 30, 26, 28, 22, 27, 21, 25, 27, 33, 42, 36, 45, 43, 46, 38, 44, 33, 43, 36, 44, 47, 60, 46, 66, 64, 70, 63, 72, 61, 69, 60, 63, 58, 69, 80

Offset: 0

David S. Newman, Jan 03 2011

nonn

Let a_1,a_2,...,a_k be a partition of n into distinct parts. We say that this partition can be refined if one of the summands, say a_i can be replaced with two numbers whose sum is a_i  and the resulting partition is a partition into distinct parts.  For example, the partition 5+2 can be refined because 5 can be replaced by 4+1 to give 4+2+1.  If a partition into distinct parts cannot be refined we say that it is maximally refined.
The value of a(0) is taken to be 1 as is often done when considering partitions (also, the empty partition cannot be refined).
This sequence was suggested by Moshe Shmuel Newman.
From Gus Wiseman, Jun 07 2025: (Start)
Given any strict partition, the following are equivalent:
1) The parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.
(End)

			a(11)=2 because there are two partitions of 11 which are maximally refined, namely 6+4+1 and 5+3+2+1.
From _Joerg Arndt_, Apr 23 2023: (Start)
The 15 maximally refined partitions of 35 are:
   1:    [ 1 2 3 4 5 6 14 ]
   2:    [ 1 2 3 4 5 7 13 ]
   3:    [ 1 2 3 4 5 8 12 ]
   4:    [ 1 2 3 4 5 9 11 ]
   5:    [ 1 2 3 4 6 7 12 ]
   6:    [ 1 2 3 4 6 8 11 ]
   7:    [ 1 2 3 4 6 9 10 ]
   8:    [ 1 2 3 4 7 8 10 ]
   9:    [ 1 2 3 5 6 7 11 ]
  10:    [ 1 2 3 5 6 8 10 ]
  11:    [ 1 2 3 5 7 8 9 ]
  12:    [ 1 2 4 5 6 7 10 ]
  13:    [ 1 2 4 5 6 8 9 ]
  14:    [ 1 3 4 5 6 7 9 ]
  15:    [ 2 3 4 5 6 7 8 ]
(End)

Massimo Lauria, Table of n, a(n) for n = 0..1500 (first 1000 terms by Fausto A. C. Cariboni)
Riccardo Aragona, Lorenzo Campioni, Roberto Civino, and Massimo Lauria, On the maximal part in unrefinable partitions of triangular numbers, arXiv:2111.11084 [math.CO], 2021.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Riccardo Aragona, Roberto Civino, Norberto Gavioli and Carlo Maria Scoppola, Unrefinable partitions into distinct parts in a normalizer chain, arXiv:2107.04666 [math.CO], 2021.
Riccardo Aragona, Lorenzo Campioni, Roberto Civino and Massimo Lauria, Verification and generation of unrefinable partitions, arXiv:2112.15096 [math.CO], 2021.
Joerg Arndt, C++ program to compute such partitions.

For subsets instead of partitions we have A326080, complement A384350.
These partitions are ranked by A383707, apparently positions of 1 in A383706.
The strict complement is A384318 (strict partitions that can be refined).
This is the strict version of A384392, ranks A384320, complement apparently A384321.
Cf. A048767, A098859, A179822, A239455, A317142, A326083, A351293, A357982, A381454, A382525, A383708, A383710, A384317.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]=={}&]],{n,0,15}] (* Gus Wiseman, Jun 09 2025 *)

More terms from Joerg Arndt, Jan 04 2011

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

A367222 Number of subsets of {1..n} whose cardinality can be written as a nonnegative linear combination of the elements.

Original entry on oeis.org

1, 2, 3, 6, 12, 24, 49, 101, 207, 422, 859, 1747, 3548, 7194, 14565, 29452, 59496, 120086, 242185, 488035, 982672, 1977166, 3975508, 7989147, 16047464, 32221270, 64674453, 129775774, 260337978, 522124197, 1046911594, 2098709858, 4206361369, 8429033614, 16887728757, 33829251009, 67755866536, 135687781793, 271693909435

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			The set {1,2,4} has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(4).
The a(0) = 1 through a(4) = 12 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {1,2}  {1,2}    {1,2}
                  {1,3}    {1,3}
                  {2,3}    {1,4}
                  {1,2,3}  {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
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.
A124506 appears to count combination-free subsets, differences of A326083.
A326020 counts complete subsets.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A008284 counts partitions by length, strict A008289.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts subsets containing two distinct elements summing to k.
Cf. A068911, A088314, A103580, A116861, A326080, A364350, A365073, A365311, A365376, A365380, A365544.

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[Length[#], Union[#]]!={}&]], {n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A367222(n):
    c, mlist = 1, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(1,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for s in mlist[k-1]:
                if s <= ws:
                    c += 1
                    break
    return c # Chai Wah Wu, Nov 16 2023

a(n) = 2^n - A367223(n).

a(13)-a(33) from Chai Wah Wu, Nov 15 2023

a(34)-a(38) from Max Alekseyev, Feb 25 2025

A367223 Number of subsets of {1..n} whose cardinality cannot be written as a nonnegative linear combination of the elements.

Original entry on oeis.org

0, 0, 1, 2, 4, 8, 15, 27, 49, 90, 165, 301, 548, 998, 1819, 3316, 6040, 10986, 19959, 36253, 65904, 119986, 218796, 399461, 729752, 1333162, 2434411, 4441954, 8097478, 14746715, 26830230, 48773790, 88605927, 160900978, 292140427, 530487359, 963610200, 1751171679, 3183997509

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

			3 cannot be written as a nonnegative linear combination of 2, 4, and 5, so {2,4,5} is counted under a(6).
The a(2) = 1 through a(6) = 15 subsets:
  {2}  {2}  {2}    {2}      {2}
       {3}  {3}    {3}      {3}
            {4}    {4}      {4}
            {3,4}  {5}      {5}
                   {3,4}    {6}
                   {3,5}    {3,4}
                   {4,5}    {3,5}
                   {2,4,5}  {3,6}
                            {4,5}
                            {4,6}
                            {5,6}
                            {2,4,5}
                            {2,4,6}
                            {2,5,6}
                            {4,5,6}

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
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.
A365046 counts combination-full subsets, differences of A364914.
Triangles:
A116861 counts positive linear combinations of strict partitions of k.
A364916 counts linear combinations of strict partitions of k.
A366320 counts subsets without a subset summing to k, with A365381.
Cf. A068911, A088314, A103580, A237667, A326020, A326080, A364350, A365073, A365312, A365377, 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[Subsets[Range[n]], combs[Length[#],Union[#]]=={}&]], {n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A367223(n):
    c, mlist = 0, []
    for m in range(1,n+1):
        t = set()
        for p in partitions(m):
            t.add(tuple(sorted(p.keys())))
        mlist.append([set(d) for d in t])
    for k in range(1,n+1):
        for w in combinations(range(1,n+1),k):
            ws = set(w)
            for s in mlist[k-1]:
                if s <= ws:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Nov 16 2023

a(n) = 2^n - A367222(n).

a(14)-a(33) from Chai Wah Wu, Nov 15 2023

a(34)-a(38) from Max Alekseyev, Feb 25 2025

A384321 Numbers whose distinct prime indices are not maximally refined.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jun 01 2025

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.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

			The prime indices of 25 are {3,3}, which has refinements: ((3),(1,2)) and ((1,2),(3)), so 25 is in the sequence.
The prime indices of 102 are {1,2,7}, which has refinement ((1),(2),(3,4)), so 102 is in the sequence.
The terms together with their prime indices begin:
     5: {3}      39: {2,6}      73: {21}
     7: {4}      41: {13}       74: {1,12}
    11: {5}      43: {14}       77: {4,5}
    13: {6}      46: {1,9}      79: {22}
    17: {7}      47: {15}       82: {1,13}
    19: {8}      49: {4,4}      83: {23}
    21: {2,4}    51: {2,7}      85: {3,7}
    22: {1,5}    53: {16}       86: {1,14}
    23: {9}      55: {3,5}      87: {2,10}
    25: {3,3}    57: {2,8}      89: {24}
    26: {1,6}    58: {1,10}     91: {4,6}
    29: {10}     59: {17}       93: {2,11}
    31: {11}     61: {18}       94: {1,15}
    33: {2,5}    62: {1,11}     95: {3,8}
    34: {1,7}    65: {3,6}      97: {25}
    35: {3,4}    67: {19}      101: {26}
    37: {12}     69: {2,9}     102: {1,2,7}
    38: {1,8}    71: {20}      103: {27}

These appear to be positions of terms > 1 in A383706, non-disjoint A357982, non-strict A299200.
The strict complement is A383707, counted by A179009.
Partitions of this type appear to be counted by A384317.
The complement is A384320.
The strict (squarefree) case appears to be A384322, counted by A384318.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A098859, A122111, A130091, A317142, A326080, A381454, A382525, A384005, A384323, A384390.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[30],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]!={}]&]

A365376 Number of subsets of {1..n} with a subset summing to n.

Original entry on oeis.org

1, 1, 2, 5, 10, 23, 47, 102, 207, 440, 890, 1847, 3730, 7648, 15400, 31332, 62922, 127234, 255374, 514269, 1030809, 2071344, 4148707, 8321937, 16660755, 33384685, 66812942, 133789638, 267685113, 535784667, 1071878216, 2144762139, 4290261840, 8583175092, 17168208940, 34342860713

Offset: 0

Gus Wiseman, Sep 08 2023

nonn

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

The case containing n is counted by A131577.
The version with re-usable parts is A365073.
The complement is counted by A365377.
The complement w/ re-usable parts is A365380.
Main diagonal of A365381.
A000009 counts sets summing to n, multisets A000041.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
A365046 counts combination-full subsets, differences of A364914.
Cf. A007865, A085489, A088809, A093971, A103580, A151897, A236912, A237668, A326080, A364534.

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

isok(s, n) = forsubset(#s, ss, if (vecsum(vector(#ss, k, s[ss[k]])) == n, return(1)));
a(n) = my(nb=0); forsubset(n, s, if (isok(s, n), nb++)); nb; \\ Michel Marcus, Sep 09 2023

from itertools import combinations, chain
from sympy.utilities.iterables import partitions
def A365376(n):
    if n == 0: return 1
    nset = set(range(1,n+1))
    s, c = [set(p) for p in partitions(n,m=n,k=n) if max(p.values(),default=1) == 1], 1
    for a in chain.from_iterable(combinations(nset,m) for m in range(2,n+1)):
        if sum(a) >= n:
            aset = set(a)
            for p in s:
                if p.issubset(aset):
                    c += 1
                    break
    return c # Chai Wah Wu, Sep 09 2023

a(n) = 2^n-A365377(n). - Chai Wah Wu, Sep 09 2023

a(16)-a(25) from Michel Marcus, Sep 09 2023
a(26)-a(32) from Chai Wah Wu, Sep 09 2023
a(33)-a(35) from Chai Wah Wu, Sep 10 2023

cp's OEIS Frontend

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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A179009 Number of maximally refined partitions of n into distinct parts.

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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.

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A367222 Number of subsets of {1..n} whose cardinality can be written as a nonnegative linear combination of the elements.

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A367223 Number of subsets of {1..n} whose cardinality cannot be written as a nonnegative linear combination of the elements.

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