A236912 -id:A236912 - cp's OEIS frontend

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.

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}]

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

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[#]]]=={}&]

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 22 2023

nonn

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

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

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

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

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

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

A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 21, 28, 35, 46, 57, 70, 87, 110, 130, 165, 198, 238, 285, 349, 410, 498, 583, 702, 819, 983, 1136, 1353, 1570, 1852, 2137, 2520, 2898, 3390, 3891, 4540, 5191, 6028, 6889, 7951, 9082, 10450, 11884, 13650, 15508, 17728, 20113

Offset: 0

Gus Wiseman, Jul 19 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 length instead of differences we have A229816, strict A240861.
For all differences of pairs parts we have A364345.
For subsets of {1..n} instead of partitions we have A364463.
The strict case is A364464.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first-differences.
Cf. A002865, A025065, A108917, A236912, A237113, A237667, A320347, A326083, A363225, A364347.

Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]=={}&]],{n,0,30}]

from collections import Counter
from sympy.utilities.iterables import partitions
def A363260(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023

cp's OEIS Frontend

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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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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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.

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

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

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