A365661 -id:A365661 - cp's OEIS frontend

A367225 Numbers m without a divisor whose prime indices sum to bigomega(m).

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 55, 58, 59, 61, 62, 63, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 106, 107, 109, 113

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

Also numbers m whose prime indices do not have a submultiset summing to bigomega(m).
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 A367213.

			The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
     3: {2}        29: {10}       58: {1,10}
     5: {3}        31: {11}       59: {17}
     7: {4}        34: {1,7}      61: {18}
    10: {1,3}      35: {3,4}      62: {1,11}
    11: {5}        37: {12}       63: {2,2,4}
    13: {6}        38: {1,8}      65: {3,6}
    14: {1,4}      41: {13}       67: {19}
    17: {7}        43: {14}       68: {1,1,7}
    19: {8}        44: {1,1,5}    71: {20}
    22: {1,5}      46: {1,9}      73: {21}
    23: {9}        47: {15}       74: {1,12}
    25: {3,3}      49: {4,4}      76: {1,1,8}
    26: {1,6}      52: {1,1,6}    77: {4,5}
    27: {2,2,2}    53: {16}       79: {22}
    28: {1,1,4}    55: {3,5}      82: {1,13}

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.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A229816 counts partitions whose length is not a part, ranks A367107.
A237667 counts sum-free partitions, ranks A364531.
A365924 counts incomplete partitions, ranks A365830.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000720, A055396, A061395, A106529, A288728, A304792, A325761, A325781, A364345, A364347.

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

A371791 Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 18, 38, 82, 175, 373, 787, 1651, 3439, 7126, 14667, 30049, 61249, 124440, 251922, 508779, 1025183, 2062287, 4142644, 8312927, 16667005, 33395275, 66880828, 133892910, 267976571, 536225921, 1072842931, 2146233971, 4293248183, 8587569636, 17176654105, 34355356676, 68713584720, 137430991937, 274867311960, 549741605972, 1099492913172, 2198998307679, 4398013970156, 8796049891377, 17592130283755, 35184298506429

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			For S = {1,3,4,6} we have {{1,6},{3,4}}, so S is counted under a(6).
The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}  {}       {}         {}         {}
              {1,2,3}  {1,2,3}    {1,2,3}    {1,2,3}
                       {1,3,4}    {1,3,4}    {1,3,4}
                       {1,2,3,4}  {1,4,5}    {1,4,5}
                                  {2,3,5}    {1,5,6}
                                  {1,2,3,4}  {2,3,5}
                                  {1,2,4,5}  {2,4,6}
                                  {2,3,4,5}  {1,2,3,4}
                                             {1,2,3,6}
                                             {1,2,4,5}
                                             {1,2,5,6}
                                             {1,3,4,6}
                                             {2,3,4,5}
                                             {2,3,5,6}
                                             {3,4,5,6}
                                             {1,2,3,4,6}
                                             {1,2,4,5,6}
                                             {2,3,4,5,6}

First differences are A232466.
The complement is counted by A371792, differences A371793.
This is the "bi-" case of A371796, differences A371797.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A357879, A365661, A366320, A365381, A365925, A367094, A371788, A371789.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],biqQ]],{n,0,15}]

a(16) onwards from Martin Fuller, Mar 21 2025

A365925 Number of subset-sums of strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 17, 22, 29, 42, 59, 74, 102, 130, 171, 226, 281, 356, 454, 566, 699, 896, 1080, 1342, 1637, 2006, 2413, 2962, 3548, 4286, 5114, 6148, 7272, 8738, 10268, 12224, 14387, 16996, 19863, 23450, 27257, 31984, 37187, 43364, 50173, 58428, 67322

Offset: 0

Gus Wiseman, Sep 26 2023

nonn

This is the "not necessarily positive" version, cf. A284640.

			The a(6) = 17 ways, showing each strict partition and its subset-sums:
    (6): 0,6
   (51): 0,1,5,6
   (42): 0,2,4,6
  (321): 0,1,2,3,4,5,6

The positive case is A284640.
The non-strict version is A304792, positive case A276024.
Row sums of A365661, non-strict A365543.
The complement (non-subset-sums) is A365922, non-strict A365918.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A365923 counts partitions by non-subset-sums, strict A365545.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A046663, A364272, A364350, A365658, A365663, A365921.

Table[Total[Length[Union[Total/@Subsets[#]]]& /@ Select[IntegerPartitions[n], UnsameQ@@#&]],{n,30}]

A367212 Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 11, 15, 22, 30, 43, 58, 80, 106, 143, 186, 248, 318, 417, 530, 684, 863, 1103, 1379, 1741, 2162, 2707, 3339, 4145, 5081, 6263, 7640, 9357, 11350, 13822, 16692, 20214, 24301, 29300, 35073, 42085, 50208, 59981, 71294, 84866, 100509, 119206

Offset: 0

Gus Wiseman, Nov 11 2023

nonn

Or, partitions whose length is a subset-sum of the parts.

			The partition (3,2,1,1) has submultisets (3,1) or (2,1,1) with sum 4, so is counted under a(7).
The a(1) = 1 through a(8) = 15 partitions:
  (1)  (11)  (21)   (22)    (32)     (42)      (52)       (62)
             (111)  (211)   (221)    (321)     (322)      (332)
                    (1111)  (311)    (2211)    (331)      (431)
                            (2111)   (3111)    (421)      (521)
                            (11111)  (21111)   (2221)     (2222)
                                     (111111)  (3211)     (3221)
                                               (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

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 partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A088809/A093971/A364534 count certain types of sum-full subsets.
A108917 counts knapsack partitions, non-knapsack A366754.
A126796 counts complete partitions, incomplete A365924.
A237668 counts sum-full partitions, sum-free A237667.
A304792 counts subset-sums of partitions, strict A365925.
Triangles:
A008284 counts partitions by length, strict A008289.
A365381 counts sets with a subset summing to k, complement A366320.
A365543 counts partitions of n with a subset-sum k, strict A365661.
Cf. A000700, A238628, A363225, A364272, A365658, A365918.

Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]

A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 12, 14, 17, 21, 25, 30, 36, 43, 51, 60, 71, 83, 97, 113, 132, 153, 178, 205, 238, 272, 315, 360, 413, 471, 539, 613, 698, 792, 899, 1018, 1153, 1302, 1470, 1658, 1867, 2100, 2362, 2652, 2974, 3335, 3734, 4178, 4672

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367224 /\ A005117.

			The strict partition (6,4,3,2,1) has submultisets {1,4} and {2,3} with sum 5 so is counted under a(16).
The a(1) = 1 through a(10) = 5 strict partitions:
  (1)  .  (2,1)  .  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                           (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                             (5,2,1)  (5,3,1)  (6,3,1)
                                                      (6,2,1)  (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.
A088809/A093971/A364534 count certain types of sum-full subsets.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A275972 counts strict knapsack partitions, non-strict A108917.
A364272 counts sum-full strict partitions, sum-free A364349.
A365925 counts subset-sums of strict partitions, non-strict A304792.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A126796, A237113, A237668, A238628, A363225, A364346, A364350, A364533, A365311, A365922.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]

A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 8, 10, 12, 15, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 95, 109, 124, 143, 162, 185, 210, 240, 270, 308, 347, 393, 443, 500, 562, 634, 711, 798, 895, 1002, 1120, 1252, 1397, 1558, 1735, 1930, 2146, 2383, 2644, 2930, 3245

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367225 /\ A005117.

			The a(2) = 1 through a(11) = 7 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (10)     (11)
            (3,1)  (4,1)  (5,1)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                                 (6,1)  (7,1)  (6,3)  (7,3)    (7,4)
                                               (8,1)  (9,1)    (8,3)
                                                      (5,4,1)  (10,1)
                                                               (5,4,2)
                                                               (6,4,1)
The a(2) = 1 through a(15) = 15 strict partitions (A..F = 10..15):
  2  3  4   5   6   7   8   9   A    B    C    D    E     F
        31  41  51  43  53  54  64   65   75   76   86    87
                    61  71  63  73   74   84   85   95    96
                            81  91   83   93   94   A4    A5
                                541  A1   B1   A3   B3    B4
                                     542  642  C1   D1    C3
                                     641  651  652  752   E1
                                          741  742  761   654
                                               751  842   762
                                               841  851   852
                                                    941   861
                                                    6521  942
                                                          951
                                                          A41
                                                          7521

Chai Wah Wu, Table of n, a(n) for n = 0..118

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.
A007865/A085489/A151897 count certain types of sum-free subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A188431 counts complete strict partitions, incomplete A365831.
A237667 counts sum-free partitions, ranks A364531.
A240861 counts strict partitions with length not a part, complement A240855.
A275972 counts strict knapsack partitions, non-strict A108917.
A364349 counts sum-free strict partitions, sum-full A364272.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365663 counts strict partitions without a subset-sum k, non-strict A046663.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A229816, A238628, A364346, A364350, A365312, A365922, A366320.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,30}]

A371796 Number of quanimous subsets of {1..n}, meaning there is more than one set partition with all equal block-sums.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 19, 43, 94, 206, 439, 946, 1990, 4204, 8761, 18233, 37778, 78151, 160296, 328670, 670193, 1363543, 2772436, 5632801, 11404156, 23071507, 46613529, 94098106, 189959349, 383407198, 773009751

Offset: 0

Gus Wiseman, Apr 17 2024

A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

			The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is counted under a(9).
The a(3) = 1 through a(6) = 19 subsets:
  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
           {1,3,4}    {1,3,4}      {1,3,4}
           {1,2,3,4}  {1,4,5}      {1,4,5}
                      {2,3,5}      {1,5,6}
                      {1,2,3,4}    {2,3,5}
                      {1,2,4,5}    {2,4,6}
                      {2,3,4,5}    {1,2,3,4}
                      {1,2,3,4,5}  {1,2,3,6}
                                   {1,2,4,5}
                                   {1,2,5,6}
                                   {1,3,4,6}
                                   {2,3,4,5}
                                   {2,3,5,6}
                                   {3,4,5,6}
                                   {1,2,3,4,5}
                                   {1,2,3,4,6}
                                   {1,2,4,5,6}
                                   {2,3,4,5,6}
                                   {1,2,3,4,5,6}

The "bi-" version for integer partitions is A002219 aerated, ranks A357976.
The "bi-" version for strict partitions is A237258 aerated, ranks A357854.
The complement for integer partitions is A321451, ranks A321453.
The version for integer partitions is A321452, ranks A321454
The version for strict partitions is A371737, complement A371736.
The complement is counted by A371789, differences A371790.
The "bi-" version is A371791, complement A371792.
First differences are A371797.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371783 counts k-quanimous partitions.
Cf. A000005, A018818, A035470, A038041, A232466, A279791, A321142, A365661, A371731, A371794, A371795.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#],SameQ@@Total/@#&]]>1&]],{n,0,10}]

a(11)-a(30) from Bert Dobbelaere, Mar 30 2025

A365918 Number of distinct non-subset-sums of integer partitions of n.

Original entry on oeis.org

0, 1, 2, 6, 8, 19, 24, 46, 60, 101, 124, 206, 250, 378, 462, 684, 812, 1165, 1380, 1927, 2268, 3108, 3606, 4862, 5648, 7474, 8576, 11307, 12886, 16652, 19050, 24420, 27584, 35225, 39604, 49920, 56370, 70540, 78608, 98419, 109666, 135212, 151176, 185875, 205308

Offset: 1

Gus Wiseman, Sep 23 2023

nonn

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The a(6) = 19 ways, showing each partition and its non-subset-sums:
       (6): 1,2,3,4,5
      (51): 2,3,4
      (42): 1,3,5
     (411): 3
      (33): 1,2,4,5
     (321):
    (3111):
     (222): 1,3,5
    (2211):
   (21111):
  (111111):

Row sums of A046663, strict A365663.
The zero-full complement (subset-sums) is A304792.
The strict case is A365922.
Weighted row-sums of A365923, rank statistic A325799, complement A365658.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A122768, A364272, A364350, A364839, A365919, A365921.

Table[Total[Length[Complement[Range[n],Total/@Subsets[#]]]&/@IntegerPartitions[n]],{n,10}]

# uses A304792_T
from sympy import npartitions
def A365918(n): return (n+1)*npartitions(n)-A304792_T(n,n,(0,),1) # Chai Wah Wu, Sep 25 2023

a(n) = (n+1)*A000041(n) - A304792(n).

a(21)-a(45) from Chai Wah Wu, Sep 25 2023

A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

Original entry on oeis.org

0, 1, 3, 6, 12, 24, 46, 90, 174, 337, 651, 1261, 2445, 4753, 9258, 18101, 35487, 69823, 137704, 272366, 539797, 1071969, 2132017, 4245964, 8464289, 16887427, 33713589, 67336900, 134542546, 268894341, 537515903, 1074640717, 2148733325, 4296686409, 8592299548, 17183084263, 34364120060, 68725368752, 137446915007, 274888501928, 549770021804, 1099530342380, 2199048203425, 4398079052052, 8796136153039, 17592241805077, 35184445671235

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The subsets of S = {1,4,6,7} have distinct sums {0,1,4,5,6,7,8,10,11,12,13,14,17,18}. Since 9 is missing, S is counted under a(7).
The a(0) = 0 through a(4) = 12 subsets:
  .  {1}  {1}    {1}    {1}
          {2}    {2}    {2}
          {1,2}  {3}    {3}
                 {1,2}  {4}
                 {1,3}  {1,2}
                 {2,3}  {1,3}
                        {1,4}
                        {2,3}
                        {2,4}
                        {3,4}
                        {1,2,4}
                        {2,3,4}

This is the "bi-" version of A371789, differences A371790.
The complement is counted by A371791, differences A232466.
First differences are A371793.
The complement is the "bi-" version of A371796, differences A371797.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371737 counts quanimous strict partitions, complement A371736.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A318434, A321455, A365543, A365661, A365663, A366320, A365381, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],Not@*biqQ]],{n,0,10}]

a(16) onwards from Martin Fuller, Mar 21 2025

A371794 Number of non-biquanimous strict integer partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 27, 23, 38, 30, 54, 43, 76, 57, 104, 79, 142, 102, 192, 138, 256, 174, 340, 232, 448, 292, 585, 375, 760, 471, 982, 602, 1260, 741, 1610, 935, 2048, 1148, 2590, 1425, 3264, 1733, 4097, 2137, 5120, 2571, 6378

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The a(1) = 1 through a(11) = 12 strict partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (521)  (81)   (91)   (92)
                                                  (432)  (631)  (A1)
                                                  (531)  (721)  (542)
                                                  (621)         (632)
                                                                (641)
                                                                (731)
                                                                (821)
                                                                (5321)

The complement is counted by A237258 aerated, ranks A357854.
Even bisection is A321142, odd A078408.
This is the "bi-" version of A371736, complement A371737.
A002219 aerated counts biquanimous partitions, ranks A357976.
A006827 and A371795 count non-biquanimous partitions, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A064914, A279787, A305551, A318434, A365543, A365663, A365661, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!biqQ[#]&]],{n,0,30}]

cp's OEIS Frontend

A367225 Numbers m without a divisor whose prime indices sum to bigomega(m).

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A371791 Number of biquanimous subsets of {1..n}. Sets with a subset having the same sum as the complement.

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A365925 Number of subset-sums of strict integer partitions of n.

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A367212 Number of integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

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A371796 Number of quanimous subsets of {1..n}, meaning there is more than one set partition with all equal block-sums.

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A365918 Number of distinct non-subset-sums of integer partitions of n.

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A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.