A336137 -id:A336137 - cp's OEIS frontend

A371795 Number of non-biquanimous integer partitions of n.

0, 1, 1, 3, 2, 7, 5, 15, 8, 30, 17, 56, 24, 101, 46, 176, 64, 297, 107, 490, 147, 792, 242, 1255, 302, 1958, 488, 3010, 629, 4565, 922, 6842, 1172, 10143, 1745, 14883, 2108, 21637, 3104, 31185, 3737, 44583, 5232, 63261, 6419, 89134, 8988, 124754, 10390, 173525

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(8) = 8 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)    (7)        (8)
            (21)   (31)  (32)     (42)   (43)       (53)
            (111)        (41)     (51)   (52)       (62)
                         (221)    (222)  (61)       (71)
                         (311)    (411)  (322)      (332)
                         (2111)          (331)      (521)
                         (11111)         (421)      (611)
                                         (511)      (5111)
                                         (2221)
                                         (3211)
                                         (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)

The complement is counted by A002219 aerated, ranks A357976.
Even bisection is A006827, odd A058695.
The strict complement is A237258, ranks A357854.
This is the "bi-" version of A321451, ranks A321453.
The complement is the "bi-" version of A321452, ranks A321454.
These partitions have ranks A371731.
The strict case is A371794, bisections A321142, A078408.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371736 counts non-quanimous strict partitons, complement A371737.
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. A035470, A064914, A305551, A336137, A365543, A365661, A365663, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n],Not@*biqQ]],{n,0,15}]

a(n) = if(n%2, numbpart(n), my(v=partitions(n/2), w=List([])); for(i=1, #v, for(j=1, i, listput(w, vecsort(concat(v[i], v[j]))))); numbpart(n)-#Set(w)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A371781 Numbers with biquanimous prime signature.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Apr 09 2024

nonn

First differs from A320911 in lacking 900.
First differs from A325259 in having 1 and lacking 120.
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 with 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 not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

A number's prime signature is given by A124010.
For prime indices we have A357976, counted by A002219 aerated.
The complement for prime indices is A371731, counted by A371795, A006827.
The complement is A371782, counted by A371840.
Partitions of this type are counted by A371839.
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.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, complement A371792.
Cf. A035470, A064914, A232466, A299702, A336137, A371737, A371796.
Subsequence of A028260.

biquanimous:= proc(L) local s,x,i,P; option remember;
  s:= convert(L,`+`); if s::odd then return false fi;
  P:= mul(1+x^i,i=L);
  coeff(P,x,s/2) > 0
end proc:
select(n -> biquanimous(ifactors(n)[2][..,2]), [$1..200]); # Robert Israel, Apr 22 2024

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; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] > 0]; q[1] = True; Select[Range[200], q] (* Amiram Eldar, Jul 24 2024 *)

A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4

Offset: 1

Gus Wiseman, Apr 13 2024

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. Sum of prime indices is given by A056239.

Factorizations into factors all having the same sum of prime indices are counted by A321455.

			The factorizations of 588 of this type are (7*7*12), (21*28), (588), so a(588) = 3.
The factorizations of 900 of this type are (5*5*6*6), (9*10*10), (25*36), (30*30), (900), so a(900) = 4.

Positions of 1's are A321453, counted by A321451.
Positions of terms > 1 are A321454, counted by A321452.
Factorizations of this type are counted by A321455, different sums A321469.
For different sums instead of same sums we have A371734.
For set partitions of binary indices we have A371735.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321142 and A371794 count non-biquanimous strict partitions.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A035470, A279787, A305551, A322794, A326515, A326518, A326534, A336137, A371783, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],SameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
A371733(n, m=n, facs=List([])) = if(1==n, if(all_have_same_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s, A371733(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

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

A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 8, 4, 1, 0, 2, 19, 24, 6, 1, 0, 2, 47, 95, 49, 9, 1, 0, 6, 105, 363, 297, 93, 12, 1, 0, 12, 248, 1292, 1660, 753, 158, 16, 1, 0, 11, 563, 4649, 8409, 5591, 1653, 250, 20, 1, 0, 2, 1414, 15976, 41264, 38074, 15590, 3249, 380, 25, 1

Offset: 0

Gus Wiseman, Apr 16 2024

			The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
Triangle begins:
     1
     0     1
     0     1     1
     0     2     2     1
     0     2     8     4     1
     0     2    19    24     6     1
     0     2    47    95    49     9     1
     0     6   105   363   297    93    12     1
     0    12   248  1292  1660   753   158    16     1
     0    11   563  4649  8409  5591  1653   250    20     1
     0     2  1414 15976 41264 38074 15590  3249   380    25     1
Row n = 4 counts the following set partitions:
  .  {{1,4},{2,3}}  {{1},{2,3,4}}    {{1},{2},{3,4}}  {{1},{2},{3},{4}}
     {{1,2,3,4}}    {{1,2},{3},{4}}  {{1},{2,3},{4}}
                    {{1,2},{3,4}}    {{1},{2,4},{3}}
                    {{1,3},{2},{4}}  {{1,4},{2},{3}}
                    {{1,3},{2,4}}
                    {{1,2,3},{4}}
                    {{1,2,4},{3}}
                    {{1,3,4},{2}}

Row sums are A000110.
Column k = 1 is A035470.
A version for integer partitions is A116608.
For block lengths instead of sums we have A208437.
A008277 counts set partitions by length.
A275780 counts set partitions with distinct block-sums.
A371737 counts quanimous strict partitions, non-strict A321452.
A371789 counts non-quanimous sets, differences A371790.
Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 3, 3

Offset: 1

Gus Wiseman, Apr 13 2024

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. Sum of prime indices is given by A056239.

Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.

			The factorizations of 90 of this type are (2*3*15), (2*5*9), (2*45), (3*30), (5*18), (6*15), (90), so a(90) = 3.

For set partitions of binary indices we have A000120, same sums A371735.
Positions of 1's are A000430.
Positions of terms > 1 are A080257.
Factorizations of this type are counted by A321469, same sums A321455.
For same instead of different sums we have A371733.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],UnsameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A371734(n, m=n, facs=List([])) = if(1==n, if(all_have_different_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s,A371734(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A371735 Maximal length of a set partition of the binary indices of n into blocks all having the same sum.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Apr 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

If a(n) = k then the binary indices of n (row n of A048793) are k-quanimous (counted by A371783).

			The binary indices of 119 are {1,2,3,5,6,7}, and the set partitions into blocks with the same sum are:
  {{1,7},{2,6},{3,5}}
  {{1,5,6},{2,3,7}}
  {{1,2,3,6},{5,7}}
  {{1,2,3,5,6,7}}
So a(119) = 3.

Set partitions of this type are counted by A035470, A336137.
A version for factorizations is A371733.
Positions of 1's are A371738.
Positions of terms > 1 are A371784.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321452 counts quanimous partitions, ranks A321454.
A326031 gives weight of the set-system with BII-number n.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A038041, A096111, A279787, A305551, A321451, A321455, A322794, A326534, A371731, A371734.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Max[Length/@Select[sps[bix[n]],SameQ@@Total/@#&]],{n,0,100}]

A336138 Number of set partitions of the binary indices of n with distinct block-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 5, 2, 4, 5, 12, 1, 2, 2, 5, 2, 5, 4, 13, 2, 4, 5, 13, 5, 13, 13, 43, 1, 2, 2, 5, 2, 5, 5, 13, 2, 5, 4, 14, 5, 13, 14, 42, 2, 4, 5, 13, 5, 14, 13, 43, 5, 13, 14, 45, 14, 44, 44, 160, 1, 2, 2, 5, 2, 5, 5, 14, 2, 5, 5, 14, 4, 13

Offset: 0

Gus Wiseman, Jul 12 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(n) set partitions for n = 3, 7, 11, 15, 23:
  {12}    {123}      {124}      {1234}        {1235}
  {1}{2}  {1}{23}    {1}{24}    {1}{234}      {1}{235}
          {13}{2}    {12}{4}    {12}{34}      {12}{35}
          {1}{2}{3}  {14}{2}    {123}{4}      {123}{5}
                     {1}{2}{4}  {124}{3}      {125}{3}
                                {13}{24}      {13}{25}
                                {134}{2}      {135}{2}
                                {1}{2}{34}    {15}{23}
                                {1}{23}{4}    {1}{2}{35}
                                {1}{24}{3}    {1}{25}{3}
                                {14}{2}{3}    {13}{2}{5}
                                {1}{2}{3}{4}  {15}{2}{3}
                                              {1}{2}{3}{5}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version for twice-partitions is A271619.
The version for partitions of partitions is (also) A271619.
These set partitions are counted by A275780.
The version for factorizations is A321469.
The version for normal multiset partitions is A326519.
The version for equal block-sums is A336137.
Set partitions with distinct block-lengths are A007837.
Set partitions of binary indices are A050315.
Twice-partitions with equal sums are A279787.
Partitions of partitions with equal sums are A305551.
Normal multiset partitions with equal block-lengths are A317583.
Multiset partitions with distinct block-sums are ranked by A326535.
Cf. A000110, A032011, A035470, A131632, A321455, A326026, A326514, A326517, A326518, A326534, A326565.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],UnsameQ@@Total/@#&]],{n,0,100}]

A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 46, 48, 50, 52, 53, 55, 56, 57, 58, 61, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 83, 84, 86, 88, 89, 91, 92

Offset: 1

Gus Wiseman, Apr 14 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 165 are {1,3,6,8}, with qualifying set partitions {{1,8},{3,6}}, and {{1,3,6,8}}, so 165 is not in the sequence.
The terms together with their binary expansions and binary indices begin:
   1:     1 ~ {1}
   2:    10 ~ {2}
   3:    11 ~ {1,2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  10:  1010 ~ {2,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  23: 10111 ~ {1,2,3,5}

Set partitions with all equal block-sums are counted by A035470.
Positions of 1's in A336137 and A371735.
The complement is A371784.
A000110 counts set partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]==1&]

A371784 Numbers with quanimous binary indices. Numbers whose binary indices can be partitioned in more than one way into blocks with the same sum.

Original entry on oeis.org

7, 13, 15, 22, 25, 27, 30, 31, 39, 42, 45, 47, 49, 51, 54, 59, 60, 62, 63, 75, 76, 82, 85, 87, 90, 93, 94, 95, 97, 99, 102, 107, 108, 109, 110, 115, 117, 119, 120, 122, 125, 126, 127, 141, 143, 147, 148, 153, 155, 158, 162, 165, 167, 170, 173, 175, 179, 180

Offset: 1

Gus Wiseman, Apr 16 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of 165 are {1,3,6,8}, with qualifying set partitions {{1,8},{3,6}}, and {{1,3,6,8}}, so 165 is in the sequence.
The terms together with their binary expansions and binary indices begin:
   7:     111 ~ {1,2,3}
  13:    1101 ~ {1,3,4}
  15:    1111 ~ {1,2,3,4}
  22:   10110 ~ {2,3,5}
  25:   11001 ~ {1,4,5}
  27:   11011 ~ {1,2,4,5}
  30:   11110 ~ {2,3,4,5}
  31:   11111 ~ {1,2,3,4,5}
  39:  100111 ~ {1,2,3,6}
  42:  101010 ~ {2,4,6}
  45:  101101 ~ {1,3,4,6}
  47:  101111 ~ {1,2,3,4,6}
  49:  110001 ~ {1,5,6}
  51:  110011 ~ {1,2,5,6}
  54:  110110 ~ {2,3,5,6}
  59:  111011 ~ {1,2,4,5,6}
  60:  111100 ~ {3,4,5,6}
  62:  111110 ~ {2,3,4,5,6}
  63:  111111 ~ {1,2,3,4,5,6}

Set partitions with all equal block-sums are counted by A035470.
Positions of terms > 1 in A336137 and A371735.
The complement is A371738.
A000110 counts set partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A001055, A006827, A038041, A305551, A321455, A326534, A371733.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Select[Range[100],Length[Select[sps[bix[#]],SameQ@@Total/@#&]]>1&]

A371732 Numbers n such that each binary index k (from row n of A048793) has the same sum of binary indices A029931(k).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 288, 512, 576, 1024, 2048, 3072, 4096, 8192, 16384, 32768, 32800, 33024, 33056, 65536, 65600, 66048, 66112, 131072, 132096, 133120, 134144, 262144, 266240, 524288, 528384, 786432, 790528, 1048576, 1056768, 2097152

Offset: 1

Gus Wiseman, Apr 13 2024

			The terms together with their binary expansions and binary indices begin:
        1:                1 ~ {1}
        2:               10 ~ {2}
        4:              100 ~ {3}
        8:             1000 ~ {4}
       12:             1100 ~ {3,4}
       16:            10000 ~ {5}
       32:           100000 ~ {6}
       64:          1000000 ~ {7}
      128:         10000000 ~ {8}
      144:         10010000 ~ {5,8}
      256:        100000000 ~ {9}
      288:        100100000 ~ {6,9}
      512:       1000000000 ~ {10}
      576:       1001000000 ~ {7,10}
     1024:      10000000000 ~ {11}
     2048:     100000000000 ~ {12}
     3072:     110000000000 ~ {11,12}
     4096:    1000000000000 ~ {13}
     8192:   10000000000000 ~ {14}
    16384:  100000000000000 ~ {15}
    32768: 1000000000000000 ~ {16}
    32800: 1000000000100000 ~ {6,16}

For prime instead of binary indices we have A326534.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A321142 and A371794 count non-biquanimous strict partitions.
A321452 counts quanimous partitions, ranks A321454.
A326031 gives weight of the set-system with BII-number n.
A357976 ranks the biquanimous partitions counted by A002219 aerated.
A371731 ranks the non-biquanimous partitions counted by A371795, A006827.
Cf. A035470, A038041, A237258, A320324, A321453, A321455, A326518, A336137, A371783, A371791, A371796.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],SameQ@@Total/@bix/@bix[#]&]

cp's OEIS Frontend

A371795 Number of non-biquanimous integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A371781 Numbers with biquanimous prime signature.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A371735 Maximal length of a set partition of the binary indices of n into blocks all having the same sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336138 Number of set partitions of the binary indices of n with distinct block-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.

Views

Author

Keywords

Comments

Examples