A237113 -id:A237113 - cp's OEIS frontend

A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 7, 12, 12, 17, 20, 26, 29, 39, 43, 54, 62, 77, 88, 107, 122, 148, 168, 200, 229, 267, 308, 360, 407, 476, 536, 623, 710, 812, 917, 1050, 1190, 1349, 1530, 1733, 1944, 2206, 2483, 2794, 3138, 3524

Offset: 0

Gus Wiseman, Aug 19 2023

nonn

			For y = (4,3,2) we can write 4 = 0*3 + 2*2, so y is counted under a(9).
For y = (11,5,3) we can write 11 = 1*5 + 2*3, so y is counted under a(19).
For y = (17,5,4,3) we can write 17 = 1*3 + 1*4 + 2*5, so y is counted under a(29).
The a(1) = 0 through a(12) = 12 strict partitions (A = 10, B = 11):
  .  .  (21)  (31)  (41)  (42)   (61)   (62)   (63)   (82)    (A1)    (84)
                          (51)   (421)  (71)   (81)   (91)    (542)   (93)
                          (321)         (431)  (432)  (532)   (632)   (A2)
                                        (521)  (531)  (541)   (641)   (B1)
                                               (621)  (631)   (731)   (642)
                                                      (721)   (821)   (651)
                                                      (4321)  (5321)  (732)
                                                                      (741)
                                                                      (831)
                                                                      (921)
                                                                      (5421)
                                                                      (6321)

For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
Non-strict versions are A364913 and the complement of A364915.
For subsets instead of partitions we have A364914, complement A326083.
The case of no all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A363226, A365002.

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@@#&&Or@@Table[combs[#[[k]], Delete[#,k]]!={}, {k,Length[#]}]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A364839(n):
    if n <= 1: return 0
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if 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

A364916 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 2, 0, 1, 0, 4, 1, 1, 3, 1, 1, 0, 5, 2, 2, 2, 3, 0, 1, 0, 6, 2, 4, 2, 3, 3, 1, 1, 0, 8, 3, 4, 4, 3, 2, 5, 0, 1, 0, 10, 3, 5, 4, 7, 4, 3, 4, 1, 1, 0, 12, 5, 6, 6, 7, 7, 4, 3, 5, 0, 1, 0, 15, 5, 9, 7, 8, 6, 12, 3, 4, 6, 1, 1, 0

Offset: 0

Gus Wiseman, Aug 17 2023

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

As a triangle, also the number of ways to write n as a *positive* linear combination of the parts of a strict integer partition of k.

			Array begins:
  1  1  1  2  2  3  4   5   6   8   10   12  15   18   22   27
  0  1  0  1  1  1  2   2   3   3   5    5   7    8    10   12
  0  1  1  2  1  2  4   4   5   6   9    10  13   15   19   23
  0  1  0  3  2  2  4   4   6   7   11   11  15   17   22   27
  0  1  1  3  3  3  7   7   8   10  16   17  23   27   33   42
  0  1  0  3  2  4  7   6   9   9   17   17  23   26   33   43
  0  1  1  5  3  4  12  10  13  16  26   27  36   42   52   68
  0  1  0  4  3  3  10  11  13  13  27   25  35   40   51   67
  0  1  1  5  4  5  15  13  19  20  36   37  51   58   72   97
  0  1  0  6  4  5  14  13  18  23  42   39  54   61   78   105
  0  1  1  6  4  6  20  17  23  25  54   50  69   80   98   138
  0  1  0  6  4  5  19  16  23  24  54   55  71   80   103  144
  0  1  1  8  6  7  27  23  30  35  72   70  103  113  139  199
  0  1  0  7  5  6  24  21  29  31  75   68  95   115  139  201
  0  1  1  8  5  7  31  27  36  39  90   86  122  137  178  255
  0  1  0  9  6  8  31  27  38  42  100  93  129  148  187  289
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   3  1  2  0  1  0
   4  1  1  3  1  1  0
   5  2  2  2  3  0  1  0
   6  2  4  2  3  3  1  1  0
   8  3  4  4  3  2  5  0  1  0
  10  3  5  4  7  4  3  4  1  1  0
  12  5  6  6  7  7  4  3  5  0  1  0
  15  5  9  7  8  6 12  3  4  6  1  1  0
  18  7 10 11 10  9 10 10  5  4  6  0  1  0
  22  8 13 11 16  9 13 11 15  5  4  6  1  1  0
  27 10 15 15 17 17 16 13 13 14  6  4  8  0  1  0

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Same as A116861 with offset 0 and rows reversed, non-strict version A364912.
Row n = 0 is A000009.
Row n = 1 is A096765.
Row n = 2 is A365005.
Column k = 0 is A000007.
Column k = 1 is A000012.
Column k = 2 is A000035.
Column k = 3 is A137719.
The main diagonal is A364910.
Left half has row sums A365002.
For not just strict partitions we have A365004, diagonal A364907.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A066328 adds up distinct prime indices.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A007865, A085489, A108917, A237113, A323092, A364272, A364533, A364670, A364911, A364913.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
t[n_,k_]:=Length[Join@@Table[combs[n,ptn],{ptn,Select[IntegerPartitions[k],UnsameQ@@#&]}]];
Table[t[k,n-k],{n,0,15},{k,0,n}]

A364345 Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 27, 34, 43, 54, 67, 83, 102, 122, 151, 182, 218, 258, 313, 366, 443, 513, 611, 713, 844, 975, 1149, 1325, 1554, 1780, 2079, 2381, 2761, 3145, 3647, 4134, 4767, 5408, 6200, 7014, 8035, 9048, 10320, 11639, 13207, 14836, 16850

Offset: 0

Gus Wiseman, Jul 20 2023

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (611)
                                               (4111)     (2222)
                                               (31111)    (3311)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

For subsets of {1..n} instead of partitions we have A007865 (sum-free sets), differences A288728.
Without re-using parts we have A236912, complement A237113.
Allowing the sum of any number of parts gives A237667 (cf. A108917).
The complement is counted by A363225, strict A363226, for subsets A093971.
The strict case is A364346.
These partitions have ranks A364347, complement A364348.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A026905, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,30}]

A364349 Number of strict integer partitions of n containing the sum of no subset of the parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 14, 21, 21, 28, 29, 38, 38, 51, 50, 65, 68, 82, 83, 108, 106, 130, 136, 163, 168, 206, 210, 248, 266, 307, 322, 381, 391, 457, 490, 553, 582, 675, 703, 797, 854, 952, 1000, 1147, 1187, 1331, 1437, 1564, 1656, 1869

Offset: 0

Gus Wiseman, Jul 29 2023

nonn

First differs from A275972 in counting (7,5,3,1), which is not knapsack.

			The partition y = (7,5,3,1) has no subset with sum in y, so is counted under a(16).
The partition y = (15,8,4,2,1) has subset {1,2,4,8} with sum in y, so is not counted under a(31).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)    (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)  (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)  (5,2)    (6,2)    (6,3)
                                        (6,1)    (7,1)    (7,2)
                                        (4,2,1)  (5,2,1)  (8,1)
                                                          (4,3,2)
                                                          (5,3,1)
                                                          (6,2,1)

For subsets of {1..n} we have A151897, complement A364534.
The non-strict version is A237667, ranked by A364531.
The complement in strict partitions is counted by A364272.
The linear combination-free version is A364350.
The binary version is A364533, allowing re-used parts A364346.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972.
A236912 counts sum-free partitions (not re-using parts), complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A085489, A093971, A111133, A240861, A320347, A325862, A363226, A364345.

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

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).

A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 10, 12, 20, 27, 39, 51, 74, 95, 130, 169, 225, 288, 378, 479, 617, 778, 990, 1239, 1560, 1938, 2419, 2986, 3696, 4538, 5575, 6810, 8319, 10102, 12274, 14834, 17932, 21587, 25963, 31120, 37275, 44513, 53097, 63181, 75092, 89030, 105460, 124647

Offset: 0

Gus Wiseman, Aug 20 2023

nonn

Includes all non-strict partitions (A047967).

			The a(0) = 0 through a(7) = 12 partitions:
  .  .  (11)  (21)   (22)    (41)     (33)      (61)
              (111)  (31)    (221)    (42)      (322)
                     (211)   (311)    (51)      (331)
                     (1111)  (2111)   (222)     (421)
                             (11111)  (321)     (511)
                                      (411)     (2221)
                                      (2211)    (3211)
                                      (3111)    (4111)
                                      (21111)   (22111)
                                      (111111)  (31111)
                                                (211111)
                                                (1111111)
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is not 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 counted under a(15).

The strict case is A364839.
For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
For subsets instead of partitions we have A364914, complement A326083.
Allowing equal parts gives A365068, complement A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A365006 = no strict partitions w/ pos linear combination.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A364346.

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@@#||Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,15}]

a(n) + A364915(n) = A000041(n).

A364346 Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 4, 5, 5, 8, 9, 11, 11, 16, 16, 20, 20, 25, 30, 34, 38, 42, 50, 58, 64, 73, 80, 90, 105, 114, 128, 148, 158, 180, 201, 220, 241, 277, 306, 333, 366, 404, 447, 497, 544, 592, 662, 708, 797, 861, 954, 1020, 1131, 1226, 1352, 1456, 1600

Offset: 0

Gus Wiseman, Jul 22 2023

nonn

			The a(1) = 1 through a(14) = 11 partitions (A..E = 10..14):
  1   2   3   4    5    6    7    8    9     A    B     C     D     E
              31   32   51   43   53   54    64   65    75    76    86
                   41        52   62   72    73   74    93    85    95
                             61   71   81    82   83    A2    94    A4
                                       531   91   92    B1    A3    B3
                                                  A1    543   B2    C2
                                                  641   732   C1    D1
                                                  731   741   652   851
                                                        831   751   932
                                                              832   941
                                                              931   A31

For subsets of {1..n} we have A007865 (sum-free sets), differences A288728.
For sums of any length > 1 we have A364349, non-strict A237667.
The complement is counted by A363226, non-strict A363225.
The non-strict version is A364345, ranks A364347, complement A364348.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions not re-using parts, complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A085489, A093971, A108917, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,15}]

from collections import Counter
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
def A364346(n): return sum(1 for p in partitions(n) if max(p.values(),default=1)==1 and not any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 20 2023

A364347 Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, Jul 26 2023

nonn

Or numbers without any prime index equal to the sum of two others, allowing re-used parts.

Also Heinz numbers of a type of sum-free partitions counted by A364345.

			We don't have 6 because prime(1), prime(1), and prime(1+1) are all divisors.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}

Subsets of this type are counted by A007865 (sum-free sets).
Partitions of this type are counted by A364345.
The squarefree case is counted by A364346.
The complement is A364348, counted by A363225.
The non-binary version is counted by A364350.
Without re-using parts we have A364461, counted by A236912.
Without re-using parts we have complement A364462, counted by A237113.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
A323092 counts double-free partitions, ranks A320340.
Cf. A093971, A237667, A288728, A325862, A326083, A363226, A364531.

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

A288728 Number of sum-free sets that can be created by adding n to all sum-free sets [1..n-1].

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 18, 19, 47, 43, 102, 116, 238, 240, 553, 554, 1185, 1259, 2578, 2607, 5873, 5526, 11834, 12601, 24692, 24390, 53735, 52534, 107445, 107330, 218727, 215607, 461367, 427778, 891039, 910294, 1804606, 1706828, 3695418, 3411513, 7136850, 6892950

Offset: 1

Ben Burns, Jun 14 2017

nonn

Using the standard definition of sum-free set, this is simply the difference of successive terms in A007865.

Number of subsets of {1..n} containing n but not containing the sum of any other two elements (repeats allowed). Also the number of sum-free sets (A007865) with maximum n. - Gus Wiseman, Aug 12 2023

			1 can be added to {};
2 can be added to {} but not {1};
3 can be added to {},{1},{2};
4 can be added to {},{1},{3} but not {2},{1,3},{2,3}.
From _Gus Wiseman_, Aug 12 2023: (Start)
The a(1) = 1 through a(7) = 18 sum-free sets with maximum n:
  {1}  {2}  {3}    {4}    {5}      {6}      {7}
            {1,3}  {1,4}  {1,5}    {1,6}    {1,7}
            {2,3}  {3,4}  {2,5}    {2,6}    {2,7}
                          {3,5}    {4,6}    {3,7}
                          {4,5}    {5,6}    {4,7}
                          {1,3,5}  {1,4,6}  {5,7}
                          {3,4,5}  {2,5,6}  {6,7}
                                   {4,5,6}  {1,3,7}
                                            {1,4,7}
                                            {1,5,7}
                                            {2,3,7}
                                            {2,6,7}
                                            {3,5,7}
                                            {4,5,7}
                                            {4,6,7}
                                            {5,6,7}
                                            {1,3,5,7}
                                            {4,5,6,7}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..88
Eric Weisstein's World of Mathematics, Sum-Free Set

Cf. A007865.
For non-binary sum-free subsets of {1..n} we have A237667.
For sum-free partitions we have A364345, without re-using parts A236912.
Without re-using parts we have A364755, diffs of A085489 (non-bin A151897).
The complement without re-using parts is A364756, differences of A088809.
Cf. A002865, A093971, A237113, A237668, A326083, A364347, A364348, A364534.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Intersection[#,Total/@Tuples[#,2]]=={}&]],{n,10}] (* Gus Wiseman, Aug 12 2023 *)

a(n) = A007865(n) - A007865(n-1).

A363226 Number of strict integer partitions of n containing some three possibly equal parts (a,b,c) such that a + b = c. A variation of sum-full strict partitions.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 5, 4, 6, 7, 11, 11, 16, 18, 26, 29, 34, 42, 51, 62, 72, 84, 101, 119, 142, 166, 191, 226, 262, 300, 354, 405, 467, 540, 623, 705, 807, 927, 1060, 1206, 1369, 1551, 1760, 1998, 2248, 2556, 2861, 3236, 3628, 4100, 4587, 5152, 5756

Offset: 0

Gus Wiseman, Jul 19 2023

nonn

Note that, by this definition, the partition (2,1) is sum-full, because (1,1,2) is a triple satisfying a + b = c.

			The a(3) = 1 through a(15) = 11 partitions (A=10, B=11, C=12):
  21  .  .  42   421  431  63   532   542   84    643   653   A5
            321       521  432  541   632   642   742   743   843
                           621  631   821   651   841   752   942
                                721   5321  921   A21   761   C21
                                4321        5421  5431  842   6432
                                            6321  6421  B21   6531
                                                  7321  5432  7431
                                                        6431  7521
                                                        6521  8421
                                                        7421  9321
                                                        8321  54321

For subsets of {1..n} we have A093971 (sum-full sets), complement A007865.
The non-strict version is A363225, ranks A364348 (complement A364347).
The complement is counted by A364346, non-strict A364345.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions not re-using parts, complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A026905, A085489, A108917, A237667, A237668 A240861, A275972, A320347, A326083.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]!={}&]],{n,0,30}]

from itertools import combinations_with_replacement
from collections import Counter
from sympy.utilities.iterables import partitions
def A363226(n): return sum(1 for p in partitions(n) if max(p.values(),default=0)==1 and any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 20 2023

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

cp's OEIS Frontend

A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

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A364916 Array read by antidiagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.

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A364345 Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.

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A364349 Number of strict integer partitions of n containing the sum of no subset of the parts.

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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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A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

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A364346 Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.

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A364347 Numbers k > 0 such that if prime(a) and prime(b) both divide k, then prime(a+b) does not.

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A288728 Number of sum-free sets that can be created by adding n to all sum-free sets [1..n-1].

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