A365043 -id:A365043 - 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

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

A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 1, 2, 6, 11, 28, 53, 118, 235, 490, 973, 2008, 3990, 8089, 16184, 32563, 65071, 130667, 261183, 523388, 1046748, 2095239, 4190208, 8385030, 16768943, 33546257, 67092732, 134201461, 268400553, 536839090, 1073670970, 2147414967, 4294829905, 8589793931

Offset: 0

Gus Wiseman, Aug 24 2023

nonn

Includes all subsets containing both 1 and n.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,4}      {1,2,5}
                        {1,2,4}    {1,3,5}
                        {1,3,4}    {1,4,5}
                        {2,3,4}    {2,3,5}
                        {1,2,3,4}  {2,4,5}
                                   {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 complement is A124506, first differences of A326083.
The binary complement is A288728, first differences of A007865.
First differences of A364914.
The positive version is A365042, first differences of A365043.
The positive complement is counted by A365045, first differences of A365044.
Without re-usable parts we have A365069, first differences of A364534.
The binary version is A365070, first differences of A093971.
A364350 counts combination-free strict partitions, complement A364839.
A085489 and A364755 count subsets without the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

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]],MemberQ[#,n]&&Or@@Table[combs[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]

a(n+1) = 2^n - A124506(n).

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

A365312 Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 2, 6, 4, 8, 7, 16, 6, 24, 17, 24, 20, 46, 22, 62, 31, 63, 57, 106, 35, 122, 90, 137, 88, 212, 74, 262, 134, 267, 206, 345, 121, 476, 294, 484, 232, 698, 242, 837, 389, 763, 571, 1185, 318, 1327, 634, 1392, 727, 1927, 640, 2056, 827, 2233, 1328

Offset: 0

Gus Wiseman, Sep 05 2023

nonn

			The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).
The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).
The a(0) = 0 through a(11) = 16 strict partitions:
  .  .  .  (2)  (3)  (2)  (4)  (2)    (3)  (2)    (3)    (2)
                     (3)  (5)  (3)    (5)  (4)    (4)    (3)
                     (4)       (4)    (6)  (5)    (6)    (4)
                               (5)    (7)  (6)    (7)    (5)
                               (6)         (7)    (8)    (6)
                               (4,2)       (8)    (9)    (7)
                                           (4,2)  (6,3)  (8)
                                           (6,2)         (9)
                                                         (10)
                                                         (4,2)
                                                         (5,4)
                                                         (6,2)
                                                         (6,3)
                                                         (6,4)
                                                         (7,3)
                                                         (8,2)

The complement for positive coefficients is counted by A088314.
For positive coefficients we have A088528.
The complement is counted by A365311.
For non-strict partitions we have A365378, complement A365379.
The version for subsets is A365380, complement A365073.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&],combs[n,#]=={}&]],{n,0,10}]

from math import isqrt
from sympy.utilities.iterables import partitions
def A365312(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n+1) for b in partitions(m,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(26)-a(58) from Chai Wah Wu, Sep 13 2023

A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 11, 12, 20, 24, 35, 38, 63, 63, 92, 112, 148, 160, 230, 244, 339, 383, 478, 533, 726, 781, 978, 1123, 1394, 1526, 1960, 2112, 2630, 2945, 3518, 3964, 4856, 5261, 6307, 7099, 8464, 9258, 11140, 12155, 14419, 16093, 18589, 20565, 24342, 26597, 30948

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

			The strict partition (6,3) cannot be linearly combined to obtain 10, so is not counted under a(10).
The strict partition (4,2) has 6 = 1*4 + 1*2 so is counted under a(6), but (4,2) cannot be linearly combined to obtain 7 so is not counted under a(7).
The a(1) = 1 through a(7) = 12 strict partitions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (2,1)  (4)    (2,1)  (3)      (2,1)
                   (2,1)  (3,1)  (6)      (3,1)
                   (3,1)  (3,2)  (2,1)    (3,2)
                          (4,1)  (3,1)    (4,1)
                                 (3,2)    (4,3)
                                 (4,1)    (5,1)
                                 (4,2)    (5,2)
                                 (5,1)    (6,1)
                                 (3,2,1)  (3,2,1)
                                          (4,2,1)

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

For positive coefficients we have A088314.
The positive complement is counted by A088528.
The version for subsets is A365073.
The complement is counted by A365312.
For non-strict partitions we have A365379.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, A365320.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions,n],UnsameQ@@#&],combs[n,#]!={}&]],{n,10}]

from math import isqrt
from sympy.utilities.iterables import partitions
def A365311(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n+1) for b in partitions(m,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(26)-a(50) from Chai Wah Wu, Sep 13 2023

A088528 Let m = number of ways of partitioning n into parts using all the parts of a subset of {1, 2, ..., n-1} whose sum of all parts of a subset is less than n; a(n) gives number of different subsets of {1, 2, ..., n-1} whose m is 0.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 6, 10, 12, 17, 18, 26, 30, 40, 44, 58, 66, 84, 95, 120, 135, 166, 186, 230, 257, 314, 350, 421, 476, 561, 626, 749, 831, 986, 1095, 1276, 1424, 1666, 1849, 2138, 2388, 2741, 3042, 3522, 3879, 4441, 4928, 5617, 6222, 7084, 7802, 8852, 9800

Offset: 1

Naohiro Nomoto, Nov 16 2003

Note that {2, 3} is counted for n = 6 because although 6 = 2+2+2 = 3+3, there is no partition that includes both 2 and 3. - David Wasserman, Aug 09 2005

Said differently, a(n) is the number of finite nonempty sets of positive integers with sum < n that cannot be linearly combined using all positive coefficients to obtain n. - Gus Wiseman, Sep 10 2023

			a(5)=3 because there are three different subsets, {2}, {3} & {4}; a(6)=3 because there are three different subsets, {4}, {5} & {2,3}.
From _Gus Wiseman_, Sep 10 2023: (Start)
The set {3,5} is not counted under a(8) because 1*3 + 1*5 = 8, but it is counted under a(9) and a(10), and it is not counted under a(11) because 2*3 + 1*5 = 11.
The a(3) = 1 through a(11) = 17 subsets:
  {2}  {3}  {2}  {4}    {2}    {3}    {2}    {3}      {2}
            {3}  {5}    {3}    {5}    {4}    {4}      {3}
            {4}  {2,3}  {4}    {6}    {5}    {6}      {4}
                        {5}    {7}    {6}    {7}      {5}
                        {6}    {2,5}  {7}    {8}      {6}
                        {2,4}  {3,4}  {8}    {9}      {7}
                                      {2,4}  {2,5}    {8}
                                      {2,6}  {2,7}    {9}
                                      {3,4}  {3,5}    {10}
                                      {3,5}  {3,6}    {2,4}
                                             {4,5}    {2,6}
                                             {2,3,4}  {2,8}
                                                      {3,6}
                                                      {3,7}
                                                      {4,5}
                                                      {4,6}
                                                      {2,3,5}
(End)

The complement is A088571, allowing sum n A088314.
For sets with max < n instead of sum < n we have A365045, nonempty A070880.
For nonnegative coefficients we have A365312, complement A365311.
For sets with max <= n we have A365322.
For partitions we have A365323, nonnegative A365378.
A116861 and A364916 count linear combinations of strict partitions.
A326083 and A124506 appear to count combination-free subsets.
Cf. A000009, A326080, A364350, A364534, A365043, A365321, A365380.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Subsets[Range[n]],0Gus Wiseman, Sep 12 2023 *)

More terms from David Wasserman, Aug 09 2005

A365322 Number of subsets of {1..n} that cannot be linearly combined using positive coefficients to obtain n.

Original entry on oeis.org

0, 1, 2, 5, 11, 26, 54, 116, 238, 490, 994, 2011, 4045, 8131, 16305, 32672, 65412, 130924, 261958, 524066, 1048301, 2096826, 4193904, 8388135, 16776641, 33553759, 67108053, 134216782, 268434324, 536869595, 1073740266, 2147481835, 4294965158, 8589932129

Offset: 0

Gus Wiseman, Sep 04 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.

			The set {1,3} has 4 = 1 + 3 so is not counted under a(4). However, 3 cannot be written as a linear combination of {1,3} using all positive coefficients, so it is counted under a(3).
The a(1) = 1 through a(4) = 11 subsets:
  {}  {}     {}       {}
      {1,2}  {2}      {3}
             {1,3}    {1,4}
             {2,3}    {2,3}
             {1,2,3}  {2,4}
                      {3,4}
                      {1,2,3}
                      {1,2,4}
                      {1,3,4}
                      {2,3,4}
                      {1,2,3,4}

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

The complement is counted by A088314.
The version for strict partitions is A088528.
The nonnegative complement is counted by A365073, without n A365542.
The binary complement is A365315, nonnegative A365314.
The binary version is A365321, nonnegative A365320.
For nonnegative coefficients we have A365380.
A085489 and A364755 count subsets without the sum of two distinct elements.
A124506 appears to count combination-free subsets, differences of A326083.
A179822 counts sum-closed subsets, first differences of A326080.
A364350 counts combination-free strict partitions, non-strict A364915.
A365046 counts combination-full subsets, first differences of A364914.
Cf. A070880, A088809, A093971, A151897, A326020, A364272, A364839, A365043, A365381.

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-> 2^n-nops(b(n$2)):
seq(a(n), n=0..33);  # Alois P. Heinz, Sep 04 2023

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

from sympy.utilities.iterables import partitions
def A365322(n): return (1<Chai Wah Wu, Sep 14 2023

a(n) = 2^n - A088314(n).

a(n) = A070880(n) + 2^(n-1) for n>=1.

More terms from Alois P. Heinz, Sep 04 2023

A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.

Original entry on oeis.org

0, 1, 1, 2, 4, 11, 23, 53, 111, 235, 483, 988, 1998, 4036, 8114, 16289, 32645, 65389, 130887, 261923, 524014, 1048251, 2096753, 4193832, 8388034, 16776544, 33553622, 67107919, 134216597, 268434140, 536869355, 1073740012, 2147481511, 4294964834, 8589931700

Offset: 0

Gus Wiseman, Aug 24 2023

nonn

Also subsets of {1..n} containing n whose greatest element cannot be written as a positive linear combination of the others.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is not counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
  .  {1}  {2}  {3}    {4}        {5}
               {2,3}  {3,4}      {2,5}
                      {2,3,4}    {3,5}
                      {1,2,3,4}  {4,5}
                                 {2,4,5}
                                 {3,4,5}
                                 {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 nonempty case is A070880.
The nonnegative version is A124506, first differences of A326083.
The binary version is A288728, first differences of A007865.
A subclass is A341507.
The complement is counted by A365042, first differences of A365043.
First differences of A365044.
The nonnegative complement is A365046, first differences of A364914.
The binary complement is A365070, first differences of A093971.
Without re-usable parts we have A365071, first differences of A151897.
A085489 and A364755 count subsets w/o the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364534, A365069.

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

a(n) = A070880(n) + 1 for n > 0.

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

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A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.

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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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A365312 Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.

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A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

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