A124506 -id:A124506 - cp's OEIS frontend

A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

1, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 7, 10, 11, 15, 17, 22, 25, 32, 37, 46, 53, 65, 75, 90, 105, 124, 143, 168, 193, 224, 258, 297, 340, 390, 446, 509, 580, 660, 751, 852, 967, 1095, 1240, 1401, 1584, 1786, 2015, 2269, 2554, 2869, 3226, 3617, 4056, 4541, 5084

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367218.

			The a(3) = 1 through a(10) = 7 strict partitions:
  (2,1)  (3,1)  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                (4,1)  (5,1)    (6,1)    (7,1)    (8,1)    (9,1)
                       (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                         (5,2,1)  (5,3,1)  (5,4,1)
                                                  (6,2,1)  (6,3,1)
                                                           (7,2,1)
                                                           (4,3,2,1)

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.
A188431 counts complete strict partitions, incomplete A365831.
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.
Cf. A008289, A088314, A116861, A124506, A363225, A364346, A364350, A364916, A365073, A365311, A365312.

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], UnsameQ@@#&&combs[Length[#], Union[#]]!={}&]], {n,0,15}]

A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 23, 24, 29, 32, 37, 41, 49, 54, 63, 72, 82, 93, 108, 122, 139, 159, 180, 204, 231, 261, 293, 331, 370, 415, 464, 518, 575, 641, 710, 789, 871, 965, 1064, 1177, 1294, 1428, 1569, 1729, 1897

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367219.

			The a(2) = 1 through a(16) = 10 strict partitions (A..G = 10..16):
  2  3  4  5  6  7   8   9   A   B    C    D    E    F    G
                 43  53  54  64  65   75   76   86   87   97
                         63  73  74   84   85   95   96   A6
                                 83   93   94   A4   A5   B5
                                 542  642  A3   B3   B4   C4
                                           652  752  C3   D3
                                           742  842  654  754
                                                     762  862
                                                     852  952
                                                     942  A42

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.
A124506 appears to count combination-free subsets, differences of A326083.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A364272 counts sum-full strict partitions, sum-free A364349.
Triangles:
A008284 counts partitions by length, strict A008289.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365541 counts subsets containing two distinct elements summing to k.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A088314, A103580, A116861, A364346, A364350, A364533, 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], UnsameQ@@#&&combs[Length[#], Union[#]]=={}&]], {n,0,30}]

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

A367227 Numbers m whose prime indices have no nonnegative linear combination equal to bigomega(m).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 63, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 117, 119, 121, 127, 131, 133, 137, 139, 143, 145, 147, 149, 151, 153, 155, 157, 161, 163

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

These are the Heinz numbers of the partitions counted by A367219.

			The prime indices of 24 are {1,1,1,2} with (1+1+1+1) = 4 or (1+1)+(2) = 4 or (2+2) = 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
     3: {2}        43: {14}        85: {3,7}
     5: {3}        47: {15}        89: {24}
     7: {4}        49: {4,4}       91: {4,6}
    11: {5}        53: {16}        95: {3,8}
    13: {6}        55: {3,5}       97: {25}
    17: {7}        59: {17}        99: {2,2,5}
    19: {8}        61: {18}       101: {26}
    23: {9}        63: {2,2,4}    103: {27}
    25: {3,3}      65: {3,6}      107: {28}
    27: {2,2,2}    67: {19}       109: {29}
    29: {10}       71: {20}       113: {30}
    31: {11}       73: {21}       115: {3,9}
    35: {3,4}      77: {4,5}      117: {2,2,6}
    37: {12}       79: {22}       119: {4,7}
    41: {13}       83: {23}       121: {5,5}

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*
A000700 counts self-conjugate partitions, ranks A088902.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124506 appears to count combination-free subsets, differences of A326083.
A229816 counts partitions whose length is not a part, ranks A367107.
A304792 counts subset-sums of partitions, strict A365925.
A365046 counts combination-full subsets, differences of A364914.
Cf. A000720, A046663, A088314, A106529, A116861, A236912, A364345, A364346, A364347, A364350, A365073, A365312.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p], {k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i}, {k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Select[Range[100], combs[PrimeOmega[#], Union[prix[#]]]=={}&]

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}]

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

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

A365320 Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 7, 5, 12, 12, 27, 14, 42, 36, 47, 47, 83, 58, 109, 80, 116, 126, 172, 111, 195, 192, 219, 202, 294, 210, 342, 286, 354, 369, 409, 324, 509, 480, 523, 452, 640, 507, 711, 622, 675, 747, 865, 654, 916, 842, 964, 922, 1124, 940, 1147, 1029

Offset: 0

Gus Wiseman, Sep 06 2023

nonn

Are there only two cases of nonzero adjacent equal parts, at positions n = 9, 15?

			The pair p = (3,6) cannot be linearly combined to obtain 8 or 10, so p is counted under a(8) and a(10), but we have 9 = 1*3 + 1*6 or 9 = 3*3 + 0*6, so p not counted under a(9).
The a(5) = 2 through a(10) = 12 pairs:
  (2,4)  (4,5)  (2,4)  (3,6)  (2,4)  (3,6)
  (3,4)         (2,6)  (3,7)  (2,6)  (3,8)
                (3,5)  (5,6)  (2,8)  (3,9)
                (3,6)  (5,7)  (4,6)  (4,7)
                (4,5)  (6,7)  (4,7)  (4,8)
                (4,6)         (4,8)  (4,9)
                (5,6)         (5,6)  (6,7)
                              (5,7)  (6,8)
                              (5,8)  (6,9)
                              (6,7)  (7,8)
                              (6,8)  (7,9)
                              (7,8)  (8,9)

The unrestricted version is A000217, ranks A001358.
For strict partitions we have A365312, complement A365311.
The (binary) complement is A365314, positive A365315.
The case of positive coefficients is A365321, for all subsets A365322.
For partitions we have A365378, complement A365379.
For all subsets instead of just pairs we have A365380, complement A365073.
A004526 counts partitions of length 2, shift right for strict.
A007865 counts sum-free subsets, complement A093971.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A070880, A088314, A088809, A151897, A326020, A364839.

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],{2}],combs[n,#]=={}&]],{n,0,30}]

from itertools import count
from sympy import divisors
def A365320(n):
    a = set()
    for i in range(1,n+1):
        if not n%i:
            a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
        else:
            for j in count(0,i):
                if j > n:
                    break
                k = n-j
                for d in divisors(k):
                    if d>=i:
                        break
                    a.add((d,i))
    return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 13 2023

cp's OEIS Frontend

A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

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A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

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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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A367227 Numbers m whose prime indices have no nonnegative linear combination equal to bigomega(m).

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

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

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

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