A365043 -id:A365043 - cp's OEIS frontend

A365314 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.

0, 0, 1, 3, 6, 8, 14, 14, 23, 24, 33, 28, 52, 36, 55, 58, 73, 53, 95, 62, 110, 94, 105, 81, 165, 105, 133, 132, 176, 112, 225, 123, 210, 174, 192, 186, 306, 157, 223, 218, 328, 180, 354, 192, 324, 315, 288, 216, 474, 260, 383, 311, 404, 254, 491, 338, 511, 360

Offset: 0

Gus Wiseman, Sep 05 2023

nonn

Is there only one case of nonzero adjacent equal parts, at position n = 6?

			We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
The a(2) = 1 through a(7) = 14 pairs:
  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)
         (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
         (2,3)  (1,4)  (1,4)  (1,4)  (1,4)
                (2,3)  (1,5)  (1,5)  (1,5)
                (2,4)  (2,3)  (1,6)  (1,6)
                (3,4)  (2,5)  (2,3)  (1,7)
                       (3,5)  (2,4)  (2,3)
                       (4,5)  (2,5)  (2,5)
                              (2,6)  (2,7)
                              (3,4)  (3,4)
                              (3,5)  (3,7)
                              (3,6)  (4,7)
                              (4,6)  (5,7)
                              (5,6)  (6,7)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

The unrestricted version is A000217, ranks A001358.
For all subsets instead of just pairs we have A365073, complement A365380.
For strict partitions we have A365311, complement A365312.
The case of positive coefficients is A365315, for all subsets A088314.
The binary complement is A365320, positive A365321.
For partitions we have A365379, complement A365378.
A004526 counts partitions of length 2, shift right for strict.
A007865 counts sum-free subsets, complement A093971.
A179822 and A326080 count sum-closed subsets.
A364350 counts combination-free strict partitions.
A364914/A365046 count combination-full subsets, complement A326083/A124506.
Cf. A070880, A088809, A151897, A326020, A365043, A365322, A365383.

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 A365314(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 len(a) # Chai Wah Wu, Sep 12 2023

A365321 Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 10, 13, 18, 24, 30, 37, 46, 54, 63, 77, 85, 99, 111, 127, 141, 161, 171, 194, 210, 235, 246, 277, 293, 322, 342, 372, 389, 428, 441, 491, 504, 545, 561, 612, 635, 680, 701, 753, 773, 836, 846, 911, 932, 1000, 1017, 1082, 1103, 1176, 1193

Offset: 0

Gus Wiseman, Sep 06 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.

			For the pair p = (2,3) we have 4 = 2*2 + 0*3, so p is not counted under A365320(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is counted under a(4).
The a(2) = 1 through a(7) = 13 pairs:
  (1,2)  (1,3)  (1,4)  (1,5)  (1,6)  (1,7)
         (2,3)  (2,3)  (2,4)  (2,3)  (2,4)
                (2,4)  (2,5)  (2,5)  (2,6)
                (3,4)  (3,4)  (2,6)  (2,7)
                       (3,5)  (3,4)  (3,5)
                       (4,5)  (3,5)  (3,6)
                              (3,6)  (3,7)
                              (4,5)  (4,5)
                              (4,6)  (4,6)
                              (5,6)  (4,7)
                                     (5,6)
                                     (5,7)
                                     (6,7)

The unrestricted version is A000217, ranks A001358.
For strict partitions we have A088528, complement A088314.
The (binary) complement is A365315, nonnegative A365314.
For nonnegative coefficients we have A365320, for subsets A365380.
For all subsets instead of just pairs we have A365322, complement A088314.
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 count combination-free subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A070880, A088571, A088809, A151897, A326020, A365043, A365073, A365311, A365312, A365378, 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[Subsets[Range[n],{2}], combp[n,#]=={}&]],{n,0,30}]

from itertools import count
from sympy import divisors
def A365321(n):
    a = set()
    for i in range(1,n+1):
        for j in count(i,i):
            if j >= n:
                break
            for d in divisors(n-j):
                if d>=i:
                    break
                a.add((d,i))
    return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 12 2023

A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 4, 4, 8, 4, 11, 9, 16, 14, 25, 20, 37, 31, 49, 47, 73, 64, 101, 96, 135, 133, 190, 181, 256, 253, 336, 342, 453, 452, 596, 609, 771, 803, 1014, 1041, 1309, 1362, 1674, 1760, 2151, 2249, 2736, 2884, 3449, 3661, 4366, 4615, 5486, 5825

Offset: 0

Gus Wiseman, Aug 31 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 a(8) = 2 through a(13) = 11 partitions:
  (8)    (9)      (10)       (11)       (12)       (13)
  (5,3)  (5,4)    (6,4)      (6,5)      (7,5)      (7,6)
         (7,2)    (7,3)      (7,4)      (5,4,3)    (8,5)
         (4,3,2)  (4,3,2,1)  (8,3)      (5,4,2,1)  (9,4)
                             (9,2)                 (10,3)
                             (5,4,2)               (11,2)
                             (6,3,2)               (6,4,3)
                             (5,3,2,1)             (6,5,2)
                                                   (7,4,2)
                                                   (5,4,3,1)
                                                   (6,4,2,1)

Chai Wah Wu, Table of n, a(n) for n = 0..109

The nonnegative version for subsets appears to be A124506.
For sums instead of combinations we have A364349, binary A364533.
The nonnegative version is A364350, complement A364839.
For subsets instead of partitions we have A365044, complement A365043.
The non-strict version is A365072, nonnegative A364915.
A000041 counts integer partitions, strict A000009.
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. A151897, A237113, A236912, A237667, A275972, A363226, A364272, A364913, A365004, A365068.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combp[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,30}]

from sympy.utilities.iterables import partitions
def A365006(n):
    if n <= 1: return 1
    alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
    c = 1
    for p in partitions(n,k=n-1):
        if max(p.values()) == 1:
            s = set(p)
            for q in s:
                if tuple(sorted(s-{q})) in alist[q]:
                    break
            else:
                c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(31)-a(56) from Chai Wah Wu, Sep 20 2023

A365044 Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

1, 2, 3, 5, 9, 20, 43, 96, 207, 442, 925, 1913, 3911, 7947, 16061, 32350, 64995, 130384, 261271, 523194, 1047208, 2095459, 4192212, 8386044, 16774078, 33550622, 67104244, 134212163, 268428760, 536862900, 1073732255, 2147472267, 4294953778, 8589918612, 17179850312

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

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

Also subsets of {1..n} such that no element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,5,6,8} has 6 = 2*3 + 0*5 + 0*8 and 8 = 1*3 + 1*5 + 0*6 but neither of these is strictly positive, so S is counted under a(8).
The a(0) = 1 through a(5) = 20 subsets:
  {}  {}   {}   {}     {}         {}
      {1}  {1}  {1}    {1}        {1}
           {2}  {2}    {2}        {2}
                {3}    {3}        {3}
                {2,3}  {4}        {4}
                       {2,3}      {5}
                       {3,4}      {2,3}
                       {2,3,4}    {2,5}
                       {1,2,3,4}  {3,4}
                                  {3,5}
                                  {4,5}
                                  {2,3,4}
                                  {2,4,5}
                                  {3,4,5}
                                  {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 version is A007865, first differences A288728.
The binary complement is A093971, first differences A365070.
Without re-usable parts we have A151897, first differences A365071.
The nonnegative version is A326083, first differences A124506.
A subclass is A341507.
The nonnegative complement is A364914, first differences A365046.
The complement is counted by A365043, first differences A365042.
First differences are A365045.
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. A006951, 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]],And@@Table[combp[Last[#],Union[Most[#]]]=={},{k,Length[#]}]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365044(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return n+1+sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] not in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365043(n).

a(15)-a(34) from Chai Wah Wu, Nov 20 2023

A365315 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 5, 8, 10, 12, 15, 18, 20, 24, 28, 28, 35, 37, 42, 44, 49, 49, 60, 59, 66, 65, 79, 74, 85, 84, 93, 93, 107, 100, 120, 104, 126, 121, 142, 129, 145, 140, 160, 150, 173, 154, 189, 170, 196, 176, 208, 193, 223, 202, 238, 203, 241, 227, 267, 235

Offset: 0

Gus Wiseman, Sep 06 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.

			We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
For the pair p = (2,3), we have 4 = 2*2 + 0*3, so p is counted under A365314(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is not counted under a(4).
The a(3) = 1 through a(10) = 15 pairs:
  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)
         (1,3)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
                (1,4)  (1,4)  (1,4)  (1,4)  (1,4)  (1,4)
                (2,3)  (1,5)  (1,5)  (1,5)  (1,5)  (1,5)
                       (2,4)  (1,6)  (1,6)  (1,6)  (1,6)
                              (2,3)  (1,7)  (1,7)  (1,7)
                              (2,5)  (2,3)  (1,8)  (1,8)
                              (3,4)  (2,4)  (2,3)  (1,9)
                                     (2,6)  (2,5)  (2,3)
                                     (3,5)  (2,7)  (2,4)
                                            (3,6)  (2,6)
                                            (4,5)  (2,8)
                                                   (3,4)
                                                   (3,7)
                                                   (4,6)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

The unrestricted version is A000217, ranks A001358.
For all subsets instead of just pairs we have A088314, complement A365322.
For strict partitions we have A088571, complement A088528.
The case of nonnegative coefficients is A365314, for all subsets A365073.
The (binary) complement is A365321, nonnegative A365320.
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, A088809, A326020, A364534, A365043, A365311, A365312, A365378, A365379, A365380, A365383.

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

from itertools import count
from sympy import divisors
def A365315(n):
    a = set()
    for i in range(1,n+1):
        for j in count(i,i):
            if j >= n:
                break
            for d in divisors(n-j):
                if d>=i:
                    break
                a.add((d,i))
    return len(a) # Chai Wah Wu, Sep 13 2023

A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 9, 11, 17, 21, 29, 36, 50, 60, 78, 95, 123, 147, 185, 221, 274, 325, 399, 472, 574, 672, 810, 945, 1131, 1316, 1557, 1812, 2137, 2462, 2892, 3322, 3881, 4460, 5176, 5916, 6846, 7817, 8993, 10250, 11765, 13333, 15280, 17308, 19731, 22306

Offset: 0

Gus Wiseman, Aug 23 2023

nonn

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

Also subsets of {1..n} containing n whose greatest element can be written as a positive linear combination of the others.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(7) = 11 subsets:
  .  .  {1,2}  {1,3}    {1,4}    {1,5}    {1,6}      {1,7}
               {1,2,3}  {2,4}    {1,2,5}  {2,6}      {1,2,7}
                        {1,2,4}  {1,3,5}  {3,6}      {1,3,7}
                        {1,3,4}  {1,4,5}  {1,2,6}    {1,4,7}
                                 {2,3,5}  {1,3,6}    {1,5,7}
                                          {1,4,6}    {1,6,7}
                                          {1,5,6}    {2,3,7}
                                          {2,4,6}    {2,5,7}
                                          {1,2,3,6}  {3,4,7}
                                                     {1,2,3,7}
                                                     {1,2,4,7}

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

The nonnegative complement is A124506, first differences of A326083.
The binary complement is A288728, first differences of A007865.
First differences of A365043.
The complement is counted by A365045, first differences of A365044.
The nonnegative version is A365046, first differences of A364914.
Without re-usable parts we have A365069, first differences of A364534.
The binary version is A365070, first differences of A093971.
A085489 and A364755 count subsets with no sum of two distinct elements.
A088314 counts sets that can be linearly combined to obtain n.
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[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combp[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]

a(n) = A088314(n) - 1.

A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 17, 15, 31, 34, 53, 65, 109, 117, 196, 224, 328, 405, 586, 673, 968, 1163, 1555, 1889, 2531, 2986, 3969, 4744, 6073, 7333, 9317, 11053, 14011, 16710, 20702, 24714, 30549, 36127, 44413, 52561, 63786, 75583, 91377, 107436, 129463

Offset: 0

Gus Wiseman, Aug 31 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 a(1) = 1 through a(8) = 6 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (11111)  (222)     (52)       (53)
                                     (111111)  (322)      (332)
                                               (1111111)  (2222)
                                                          (11111111)
The a(11) = 17 partitions:
  (11)  (9,2)  (7,2,2)  (5,3,2,1)  (4,3,2,1,1)  (1,1,1,1,1,1,1,1,1,1,1)
        (8,3)  (6,3,2)  (5,2,2,2)  (3,2,2,2,2)
        (7,4)  (5,4,2)  (4,3,2,2)
        (6,5)  (5,3,3)  (3,3,3,2)
               (4,4,3)

The nonnegative version is A364915, strict A364350.
The strict case is A365006.
For subsets instead of partitions we have A365044, complement A365043.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A237667 counts sum-free partitions, binary A236912.
A364912 counts positive linear combinations of partitions.
A365068 counts combination-full partitions, strict A364839.
Cf. A085489, A108917, A151897, A325862, A364272, A364910, A364911, A364913.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Union/@IntegerPartitions[n], Function[ptn,!Or@@Table[combp[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A365072(n):
    if n <= 1: return 1
    alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
    c = 1
    for p in partitions(n,k=n-1):
        s = set(p)
        for q in s:
            if tuple(sorted(s-{q})) in alist[q]:
                break
        else:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(31)-a(49) from Chai Wah Wu, Sep 20 2023

cp's OEIS Frontend

A365314 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.

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A365321 Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.

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A365006 Number of strict integer partitions of n such that no part can be written as a (strictly) positive linear combination of the others.

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A365044 Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.

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A365315 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.

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A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

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A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.

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