A151897 -id:A151897 - cp's OEIS frontend

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

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

A364915 Number of integer partitions of n such that no distinct part can be written as a nonnegative linear combination of other distinct parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 12, 10, 16, 16, 19, 21, 29, 25, 37, 35, 44, 46, 60, 55, 75, 71, 90, 90, 114, 110, 140, 138, 167, 163, 217, 201, 248, 241, 298, 303, 359, 355, 425, 422, 520, 496, 594, 603, 715, 706, 834, 826, 968, 972, 1153, 1147, 1334, 1315, 1530

Offset: 0

Gus Wiseman, Aug 22 2023

nonn

			The a(1) = 1 through a(10) = 8 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    32     33      43       44        54         55
              1111  11111  222     52       53        72         64
                           111111  322      332       333        73
                                   1111111  2222      522        433
                                            11111111  3222       3322
                                                      111111111  22222
                                                                 1111111111
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is counted under a(12).
The partition (6,4,3,2) has 6=4+2, or 6=3+3, or 6=2+2+2, or 4=2+2, so is not counted under a(15).

For sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The strict case is A364350.
The complement is A365068, strict A364839.
The positive case is A365072, strict A365006.
A000041 counts integer partitions, strict A000009.
A007865 counts binary sum-free sets w/ re-usable parts, complement A093971.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364912 counts linear combinations of partitions of k.
Cf. A085489, A108917, A151897, A237113, A323092, A364272, A364533, A364910, A364911, A364913.

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], Function[ptn,!Or@@Table[combs[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]], {n,0,15}]

from sympy.utilities.iterables import partitions
def A364915(n):
    if n <= 1: return 1
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
    for p in partitions(n,k=n-1):
        s = set(p)
        if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
            c += 1
    return c # Chai Wah Wu, Sep 23 2023

a(n) = A000041(n) - A365068(n).

a(37)-a(59) from Chai Wah Wu, Sep 25 2023

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

A366320 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} without a subset summing to k.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 3, 6, 6, 7, 8, 8, 6, 6, 9, 11, 11, 14, 14, 15, 16, 16, 12, 12, 9, 17, 17, 20, 20, 24, 27, 27, 30, 30, 31, 32, 32, 24, 24, 18, 17, 26, 31, 29, 35, 36, 43, 47, 50, 51, 56, 59, 59, 62, 62, 63

Offset: 1

Gus Wiseman, Oct 12 2023

			Triangle begins:
   1
   2  2  3
   4  4  3  6  6  7
   8  8  6  6  9 11 11 14 14 15
  16 16 12 12  9 17 17 20 20 24 27 27 30 30 31
  32 32 24 24 18 17 26 31 29 35 36 43 47 50 51 56 59 59 62 62 63
Row n = 3 counts the following subsets:
  {}     {}     {}   {}     {}     {}
  {2}    {1}    {1}  {1}    {1}    {1}
  {3}    {3}    {2}  {2}    {2}    {2}
  {2,3}  {1,3}       {3}    {3}    {3}
                     {1,2}  {1,2}  {1,2}
                     {2,3}  {1,3}  {1,3}
                                   {2,3}

Row lengths are A000217.
The diagonal T(n,n) is A365377, complement A365376.
The complement is counted by A365381.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 counts combination-free subsets, differences of A326083.
A365046 counts combination-full subsets, differences of A364914.
Cf. A007865, A085489, A093971, A103580, A151897, A326080, A364534, A365073, A365380.

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

A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Jul 27 2023

nonn

Also Heinz numbers of a type of sum-free partitions not allowing re-used parts, counted by A236912.

			The prime indices of 198 are {1,2,2,5}, which is sum-free even though it is not knapsack (A299702, A299729), so 198 is in the sequence.

Subsets of this type are counted by A085489, with re-usable parts A007865.
Subsets not of this type are counted by A093971, w/ re-usable parts A088809.
Partitions of this type are counted by A236912.
Allowing parts to be re-used gives A364347, counted by A364345.
The complement allowing parts to be re-used is A364348, counted by A363225.
The non-binary version allowing re-used parts is counted by A364350.
The complement is A364462, counted by A237113.
The non-binary version is A364531, counted by A237667, complement A364532.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A288728, A320340, A325862, A325864, A326083, A363226, A364346.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]=={}&]

A364462 Positive integers having a divisor of the form prime(a)*prime(b) such that prime(a+b) is also a divisor.

Original entry on oeis.org

12, 24, 30, 36, 48, 60, 63, 70, 72, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 154, 156, 165, 168, 180, 189, 192, 204, 210, 216, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 324, 325, 330, 336, 348, 350, 360, 372, 378, 384, 390

Offset: 1

Gus Wiseman, Jul 29 2023

nonn

Also Heinz numbers of a type of sum-full partitions not allowing re-used parts, counted by A237113.
No partitions of this type are knapsack (A299702, A299729).
All multiples of terms are terms. - Robert Israel, Aug 30 2023

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   72: {1,1,1,2,2}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}

Robert Israel, Table of n, a(n) for n = 1..10000

Subsets not of this type are counted by A085489, w/ re-usable parts A007865.
Subsets of this type are counted by A088809, with re-usable parts A093971.
Partitions not of this type are counted by A236912.
Partitions of this type are counted by A237113.
Subset of A299729.
The complement with re-usable parts is A364347, counted by A364345.
With re-usable parts we have A364348, counted by A363225 (strict A363226).
The complement is A364461.
The non-binary complement is A364531, counted by A237667.
The non-binary version is A364532, see also A364350.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A320340, A325862, A326083.

filter:= proc(n) local F, i,j,m;
  F:= map(t -> `if`(t[2]>=2, numtheory:-pi(t[1])$2, numtheory:-pi(t[1])), ifactors(n)[2]);
  for i from 1 to nops(F)-1 do for j from 1 to i-1 do
    if member(F[i]+F[j],F) then return true fi
  od od;
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 30 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]!={}&]

A365043 Number of subsets of {1..n} whose greatest element can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 21, 32, 49, 70, 99, 135, 185, 245, 323, 418, 541, 688, 873, 1094, 1368, 1693, 2092, 2564, 3138, 3810, 4620, 5565, 6696, 8012, 9569, 11381, 13518, 15980, 18872, 22194, 26075, 30535, 35711, 41627, 48473, 56290, 65283, 75533, 87298, 100631, 115911, 133219

Offset: 0

Gus Wiseman, Aug 25 2023

nonn

Sets of this type may be called "positive combination-full".

Also subsets of {1..n} such that some element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,4,9} has 9 = 3*3 + 0*4, but this is not strictly positive, so S is not counted under a(9).
The subset S = {3,4,10} has 10 = 2*3 + 1*4, so S is counted under a(10).
The a(0) = 0 through a(5) = 12 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}
                                 {1,2,5}
                                 {1,3,4}
                                 {1,3,5}
                                 {1,4,5}
                                 {2,3,5}

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

The binary complement is A007865, first differences A288728.
The binary version is A093971, first differences A365070.
The nonnegative complement is A326083, first differences A124506.
The nonnegative version is A364914, first differences A365046.
First differences are A365042.
The complement is counted by A365044, first differences A365045.
Without re-usable parts we have A364534, first differences A365069.
A085489 and A364755 count subsets with no sum of two distinct elements.
A088809 and A364756 count subsets with some sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Rest[Subsets[Range[n]]],combp[Last[#],Union[Most[#]]]!={}&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365043(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365044(n).

a(15)-a(35) from Chai Wah Wu, Nov 20 2023

More terms from Bert Dobbelaere, Apr 28 2025

A364533 Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 15, 21, 22, 28, 32, 38, 40, 51, 55, 65, 74, 83, 94, 111, 119, 136, 160, 174, 196, 222, 252, 273, 315, 341, 391, 425, 477, 518, 602, 636, 719, 782, 886, 944, 1073, 1140, 1302, 1380, 1553, 1651, 1888, 1995, 2224, 2370

Offset: 0

Gus Wiseman, Aug 02 2023

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4    5    6    7     8     9     A     B     C
          21   31   32   42   43    53    54    64    65    75
                    41   51   52    62    63    73    74    84
                              61    71    72    82    83    93
                              421   521   81    91    92    A2
                                          432   631   A1    B1
                                          531   721   542   543
                                          621         632   732
                                                      641   741
                                                      731   831
                                                      821   921

For subsets of {1..n} we have A085489, complement A088809.
The non-strict version is A236912, complement A237113, ranked by A364461.
Allowing re-used parts gives A364346.
The non-binary version is A364349, non-strict A237667 (complement A237668).
The linear combination-free version is A364350.
The complement in strict partitions is A364670, w/ re-used parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972.
A151897 counts sum-free subsets, complement A364534.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A240861, A325862, A326083, A364272, A364345, A364347.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2}]] == {}&]],{n,0,30}]

A365377 Number of subsets of {1..n} without a subset summing to n.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 17, 26, 49, 72, 134, 201, 366, 544, 984, 1436, 2614, 3838, 6770, 10019, 17767, 25808, 45597, 66671, 116461, 169747, 295922, 428090, 750343, 1086245, 1863608, 2721509, 4705456, 6759500, 11660244, 16877655, 28879255, 41778027, 71384579, 102527811, 176151979

Offset: 0

Gus Wiseman, Sep 08 2023

nonn

			The a(1) = 1 through a(6) = 17 subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {1,2}  {4}    {4}
                {2,3}  {1,2}  {5}
                       {1,3}  {1,2}
                       {2,4}  {1,3}
                       {3,4}  {1,4}
                              {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {1,3,4}
                              {2,3,5}
                              {3,4,5}

David A. Corneth, Table of n, a(n) for n = 0..60

The complement w/ re-usable parts is A365073.
The complement is counted by A365376.
The version with re-usable parts is A365380.
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.
A365381 counts subsets of {1..n} with a subset summing to k.
Cf. A007865, A085489, A088809, A093971, A103580, A151897, A236912, A237113, A237668, A326080, A364349, A364534.

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

isok(s, n) = forsubset(#s, ss, if (vecsum(vector(#ss, k, s[ss[k]])) == n, return(0))); 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 A365377(n):
    if n == 0: return 0
    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 (1<Chai Wah Wu, Sep 09 2023

a(n) = 2^n-A365376(n). - Chai Wah Wu, Sep 09 2023

a(16)-a(27) from Michel Marcus, Sep 09 2023
a(28)-a(32) from Chai Wah Wu, Sep 09 2023
a(33)-a(35) from Chai Wah Wu, Sep 10 2023
More terms from David A. Corneth, Sep 10 2023

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
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.
These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Partitions not of this type are counted by A237667, strict A364349.
Partitions of this type are counted by A237668, strict A364272.
The binary complement is A364461, re-usable A364347 (counted by A364345).
The binary version is A364462, re-usable A364348 (counted by A363225).
The complement is A364531.
Subsets of this type are counted by A364534, complement A151897.
A000005 counts divisors, nonprime A033273, composite A055212.
A001222 counts prime indices.
A108917 counts knapsack partitions, strict A275972, for subsets A325864.
A112798 lists prime indices, sum A056239.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, complement A299729.
Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

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