A237258 -id:A237258 - cp's OEIS frontend

A371782 Numbers with non-biquanimous prime signature.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 102

Offset: 1

Gus Wiseman, Apr 09 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 (aerated) and ranked by A357976.

Also numbers n without a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.

			The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

A number's prime signature is given by A124010.
The complement for prime indices is A357976, counted by A002219 aerated.
For prime indices we have A371731, counted by A371795, even case A006827.
The complement is A371781, counted by A371839.
Partitions of this type are counted by A371840.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
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.
A371792 counts non-biquanimous sets, complement A371791.
Cf. A035470, A064914, A232466, A299729, A367094, A371736, A371783, A371789.
Subsequence of A026424.

g[n_]:=Select[Divisors[n],GCD[#,n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100],g[#]=={}&]
(* second program: *)
q[n_] := Module[{e = FactorInteger[n][[;; , 2]], sum, x}, sum = Plus @@ e; OddQ[sum] || CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] == 0]; q[1] = False; Select[Range[120], q] (* Amiram Eldar, Jul 24 2024 *)

A371736 Number of non-quanimous strict integer partitions of n, meaning no set partition with more than one block has all equal block-sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 12, 11, 18, 15, 26, 23, 38, 30, 54, 43, 72, 57, 104, 77, 142, 102, 179, 138, 256, 170, 340, 232, 412, 292, 585, 365, 760, 471, 889, 602, 1260, 718, 1610, 935, 1819, 1148, 2590, 1371, 3264, 1733, 3581, 2137, 5120, 2485, 6372

Offset: 0

Gus Wiseman, Apr 14 2024

nonn

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 a(0) = 1 through a(9) = 8 strict partitions:
  ()  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
                (21)  (31)  (32)  (42)  (43)   (53)   (54)
                            (41)  (51)  (52)   (62)   (63)
                                        (61)   (71)   (72)
                                        (421)  (521)  (81)
                                                      (432)
                                                      (531)
                                                      (621)

The non-strict "bi-" complement is A002219, ranks A357976.
The "bi-" version is A321142 or A371794, complement A237258, ranks A357854.
The non-strict version is A321451, ranks A321453.
The complement is A371737, non-strict A321452, ranks A321454.
The non-strict "bi-" version is A371795, ranks A371731.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371792 counts non-biquanimous sets, complement A371791.
A371796 counts quanimous sets, differences A371797.
Cf. A000005, A018818, A035470, A038041, A232466, A279791, A365663, A365661, A365925, A371840.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Select[sps[#], SameQ@@Total/@#&]]==1&]],{n,0,30}]

a(prime(k)) = A064688(k) = A000009(A000040(k)).

A371793 Number of non-biquanimous subsets of {1..n} containing n.

Original entry on oeis.org

1, 2, 3, 6, 12, 22, 44, 84, 163, 314, 610, 1184, 2308, 4505, 8843, 17386, 34336, 67881, 134662, 267431, 532172, 1060048, 2113947, 4218325, 8423138, 16826162, 33623311, 67205646, 134351795, 268621562, 537124814, 1074092608, 2147953084, 4295613139, 8590784715, 17181035797, 34361248692, 68721546255, 137441586921, 274881519876, 549760320576, 1099517861045, 2199030848627, 4398057100987, 8796105652038, 17592203866158

Offset: 1

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(5) = 12 subsets:
  {1}  {2}    {3}    {4}      {5}
       {1,2}  {1,3}  {1,4}    {1,5}
              {2,3}  {2,4}    {2,5}
                     {3,4}    {3,5}
                     {1,2,4}  {4,5}
                     {2,3,4}  {1,2,5}
                              {1,3,5}
                              {2,4,5}
                              {3,4,5}
                              {1,2,3,5}
                              {1,3,4,5}
                              {1,2,3,4,5}

The complement is counted by A232466, differences of A371791.
This is the "bi-" version of A371790, differences of A371789.
First differences of A371792.
The complement is the "bi-" version of A371797, differences of A371796.
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, A365543, A365661, A365663, A366320, A365381, A367094, A371788.

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

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

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

Offset: 0

Wouter Meeussen, Jul 09 2005

nonn

Counted as orderless pairs since intersection is commutative.

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)

a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Name edited by Gus Wiseman, Oct 10 2023

a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Oct 27 2022

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.

			The a(3600) = 5 divisors, their prime indices, and the prime indices of their quotients:
  45: {2,2,3} * {1,1,1,1,3}
  50: {1,3,3} * {1,1,1,2,2}
  60: {1,1,2,3} * {1,1,2,3}
  72: {1,1,1,2,2} * {1,3,3}
  80: {1,1,1,1,3} * {2,2,3}

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of nonzero terms are A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A213074, A235130, A237258, A276107, A300061, A321144, A357975.

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Table[Length[Select[Divisors[n],sumprix[#]==sumprix[n]/2&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
A357879(n) = sumdiv(n,d, A056239(d)==A056239(n/d)); \\ Antti Karttunen, Jan 20 2025

a(n) = Sum_{d|n} [A056239(d) = A056239(n/d)], where [ ] is the Iverson bracket. - Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A365662 Number of ordered pairs of disjoint strict integer partitions of n.

Original entry on oeis.org

1, 0, 0, 2, 2, 6, 8, 14, 18, 32, 42, 66, 92, 136, 190, 280, 374, 532, 744, 1014, 1366, 1896, 2512, 3384, 4526, 6006, 7910, 10496, 13648, 17842, 23338, 30116, 38826, 50256, 64298, 82258, 105156, 133480, 169392, 214778, 270620, 340554, 428772, 536302, 670522

Offset: 0

Gus Wiseman, Sep 19 2023

nonn

Also the number of ways to first choose a strict partition of 2n, then a subset of it summing to n.

			The a(0) = 1 through a(7) = 14 pairs:
  ()()  .  .  (21)(3)  (31)(4)  (32)(5)   (42)(6)   (43)(7)
              (3)(21)  (4)(31)  (41)(5)   (51)(6)   (52)(7)
                                (5)(32)   (6)(42)   (61)(7)
                                (5)(41)   (6)(51)   (7)(43)
                                (32)(41)  (321)(6)  (7)(52)
                                (41)(32)  (42)(51)  (7)(61)
                                          (51)(42)  (421)(7)
                                          (6)(321)  (43)(52)
                                                    (43)(61)
                                                    (52)(43)
                                                    (52)(61)
                                                    (61)(43)
                                                    (61)(52)
                                                    (7)(421)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

For subsets instead of partitions we have A000244, non-disjoint A000302.
If the partitions can have different sums we get A032302.
The non-strict version is A054440, non-disjoint A001255.
The unordered version is A108796, non-strict A260669.
A000041 counts integer partitions, strict A000009.
A000124 counts distinct possible sums of subsets of {1..n}.
A000712 counts distinct submultisets of partitions.
A002219 and A237258 count partitions of 2n including a partition of n.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A006827, A046663, A064914, A122768, A260664, A365661, A365663.

Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2], Intersection@@#=={}&]], {n,0,15}]
Table[SeriesCoefficient[Product[(1 + x^k + y^k), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Apr 24 2025 *)

a(n) = 2*A108796(n) for n > 1.

a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k). - Ilya Gutkovskiy, Apr 24 2025

A002220 a(n) is the number of partitions of 3n that can be obtained by adding together three (not necessarily distinct) partitions of n.

Original entry on oeis.org

1, 4, 10, 30, 65, 173, 343, 778, 1518, 3088, 5609, 10959, 18990, 34441, 58903, 102044, 167499, 282519, 451529, 737208, 1160102, 1836910, 2828466, 4410990, 6670202, 10161240, 15186315, 22758131, 33480869

Offset: 1

N. J. A. Sloane

nonn

			From _Gus Wiseman_, Apr 20 2024: (Start)
The a(1) = 1 through a(3) = 10 triquanimous partitions:
  (111)  (222)     (333)
         (2211)    (3321)
         (21111)   (32211)
         (111111)  (33111)
                   (222111)
                   (321111)
                   (2211111)
                   (3111111)
                   (21111111)
                   (111111111)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.

See A002219 for further details. Cf. A002221, A002222, A213074.
A column of A213086.
For biquanimous we have A002219, ranks A357976.
For non-biquanimous we have A371795, ranks A371731, even case A006827.
The Heinz numbers of these partitions are given by A371955.
The strict case is A372122.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A237258, A321142, A371737, A371792, A371796.

Edited by N. J. A. Sloane, Jun 03 2012
a(12)-a(20) from Alois P. Heinz, Jul 10 2012
a(21)-a(29) from Sean A. Irvine, Sep 05 2013

A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 05 2014

			First 13 rows:
1
0 1
1 1 2
1 0 1 2
1 1 1 1 3
2 2 1 2 2 4
2 2 2 2 2 2 5
3 2 3 1 3 2 3 6
3 3 4 3 3 4 3 3 8
5 4 5 4 3 4 5 4 5 10
5 5 5 5 5 5 5 5 5 5 12
7 6 7 7 7 4 7 7 7 6 7 15
8 7 8 8 8 8 8 8 8 8 7 8 18
T(12,4) = 7 counts these partitions:  [8,4], [8,3,1], [7,4,1], [6,4,2], [6,3,2,1], [5,4,3], [5,4,2,1].

Clark Kimberling, Table of n, a(n) for n = 1..1000

Column k = n is A000009.
Column k = 2 is A015744.
Column k = 1 is A025147.
The non-strict complement is obtained by adding zeros after A046663.
Diagonal n = 2k is A237258.
Row sums are A284640.
For subsets instead of partitions we have A365381.
The non-strict version is obtained by removing column k = 0 from A365543.
Including column k = 0 gives A365661.
The complement is obtained by adding zeros after A365663.
Cf. A002219, A006827, A108796, A108917, A122768, A275972, A299701, A304792, A364272.

Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[nn], # == DeleteDuplicates[#] &]]; Table[Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], sumTo]] >= 1 &]]], {sumTo, nn}], {nn, 45}] // TableForm
u = Flatten[%]  (* Peter J. C. Moses, Feb 04 2014 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n,6}, {k,n}] (* Gus Wiseman, Nov 16 2023 *)

T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.

A371839 Number of integer partitions of n with biquanimous multiplicities.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 4, 6, 9, 11, 16, 22, 29, 38, 52, 66, 88, 114, 147, 186, 245, 302, 389, 486, 613, 757, 960, 1172, 1466, 1790, 2220, 2695, 3332, 4013, 4926, 5938, 7228, 8660, 10519, 12545, 15151, 18041, 21663, 25701, 30774, 36361, 43359, 51149, 60720, 71374

Offset: 0

Gus Wiseman, Apr 18 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 partition y = (6,2,1,1) has multiplicities (1,1,2), which are biquanimous because we have the partition ((1,1),(2)), so y is counted under a(10).
The a(0) = 1 through a(10) = 11 partitions:
  ()  .  .  (21)  (31)  (32)  (42)    (43)    (53)    (54)      (64)
                        (41)  (51)    (52)    (62)    (63)      (73)
                              (2211)  (61)    (71)    (72)      (82)
                                      (3211)  (3221)  (81)      (91)
                                              (3311)  (3321)    (3322)
                                              (4211)  (4221)    (4321)
                                                      (4311)    (4411)
                                                      (5211)    (5221)
                                                      (222111)  (5311)
                                                                (6211)
                                                                (322111)

For parts instead of multiplicities we have A002219 aerated, ranks A357976.
These partitions have Heinz numbers A371781.
The complement for parts instead of multiplicities is counted by A371795, ranks A371731, bisections A006827, A058695.
The complement is counted by A371840, ranks A371782.
A237258 = biquanimous strict partitions, ranks A357854, complement A371794.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A035470, A064914, A305551, A321142, A365543, A365925, A367094.

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

A371840 Number of integer partitions of n with non-biquanimous multiplicities.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 8, 11, 16, 21, 31, 40, 55, 72, 97, 124, 165, 209, 271, 343, 441, 547, 700, 866, 1089, 1345, 1679, 2050, 2546, 3099, 3814, 4622, 5654, 6811, 8297, 9957, 12039, 14409, 17355, 20666, 24793, 29432, 35133, 41598, 49474, 58360, 69197, 81395, 96124

Offset: 0

Gus Wiseman, Apr 18 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 partition y = (6,2,1,1) has multiplicities (1,1,2), which are biquanimous because we have the partition ((1,1),(2)), so y is not counted under a(10).
The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (321)     (421)      (422)
                            (11111)  (411)     (511)      (431)
                                     (3111)    (2221)     (521)
                                     (21111)   (4111)     (611)
                                     (111111)  (22111)    (2222)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

The complement for parts is counted by A002219 aerated, ranks A357976.
These partitions have Heinz numbers A371782.
For parts we have A371795, ranks A371731, bisections A006827, A058695.
The complement is counted by A371839, ranks A371781.
A237258 = biquanimous strict partitions, ranks A357854, complement A371794.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A035470, A064914, A305551, A321142, A365543, A365925, A367094.

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

cp's OEIS Frontend

A371782 Numbers with non-biquanimous prime signature.

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A371736 Number of non-quanimous strict integer partitions of n, meaning no set partition with more than one block has all equal block-sums.

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A371793 Number of non-biquanimous subsets of {1..n} containing n.

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A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

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A357879 Number of divisors of n with the same sum of prime indices as their quotient. Central column of A321144, taking gaps as 0's.

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A365662 Number of ordered pairs of disjoint strict integer partitions of n.

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A002220 a(n) is the number of partitions of 3n that can be obtained by adding together three (not necessarily distinct) partitions of n.

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A237194 Triangular array: T(n,k) = number of strict partitions P of n into positive parts such that P includes a partition of k.

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