A371783 -id:A371783 - cp's OEIS frontend

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

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}]

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}]

A371955 Numbers with triquanimous prime indices.

Original entry on oeis.org

8, 27, 36, 48, 64, 125, 150, 180, 200, 216, 240, 288, 320, 343, 384, 441, 490, 512, 567, 588, 630, 700, 729, 756, 784, 810, 840, 900, 972, 1000, 1008, 1080, 1120, 1200, 1296, 1331, 1344, 1440, 1600, 1694, 1728, 1792, 1815, 1920, 2156, 2178, 2197, 2304, 2310

Offset: 1

Gus Wiseman, Apr 19 2024

nonn

A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums.

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 terms together with their prime indices begin:
     8: {1,1,1}
    27: {2,2,2}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
   125: {3,3,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   200: {1,1,1,3,3}
   216: {1,1,1,2,2,2}
   240: {1,1,1,1,2,3}
   288: {1,1,1,1,1,2,2}
   320: {1,1,1,1,1,1,3}
   343: {4,4,4}
   384: {1,1,1,1,1,1,1,2}
   441: {2,2,4,4}
   490: {1,3,4,4}
   512: {1,1,1,1,1,1,1,1,1}
   567: {2,2,2,2,4}
   588: {1,1,2,4,4}

Robert Israel, Table of n, a(n) for n = 1..500

These are the Heinz numbers of the partitions counted by A002220.
For biquanimous we have A357976, counted by A002219.
For non-biquanimous we have A371731, counted by A371795, even case A006827.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A371783 counts k-quanimous partitions.
Cf. A000005, A002221, A035470, A064914, A232466, A299701, A321142, A321454, A321455, A326534, A357879, A371781.

tripart:= proc(L) local t,X,Y,n,cons,i,R;
  t:= convert(L,`+`)/3;
  n:= nops(L);
  if not t::integer then return false fi;
  cons:= [add(L[i]*X[i],i=1..n)=t,
          add(L[i]*Y[i],i=1..n)=t,
          seq(X[i] + Y[i] <= 1, i=1..n)];
  R:= traperror(Optimization:-Maximize(0, cons, assume=binary));
  R::list
end proc:
primeindices:= proc(n) local F,t;
  F:= ifactors(n)[2];
  map(t -> numtheory:-pi(t[1])$t[2], F)
end proc:
select(tripart @ primindices, [$2..3000]); # Robert Israel, May 19 2025

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]

A371956 Number of non-biquanimous compositions of 2n.

Original entry on oeis.org

0, 1, 3, 9, 23, 63, 146, 364

Offset: 0

Gus Wiseman, Apr 20 2024

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(3) = 9 compositions:
  (2)  (4)    (6)
       (1,3)  (1,5)
       (3,1)  (2,4)
              (4,2)
              (5,1)
              (1,1,4)
              (1,4,1)
              (2,2,2)
              (4,1,1)

The unordered complement is A002219, ranks A357976.
The unordered version is A006827, even case of A371795, ranks A371731.
The complement is counted by A064914.
These compositions have ranks A372119, complement A372120.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2n], !MemberQ[Total/@Subsets[#],n]&]],{n,0,5}]

A372119 Numbers k such that the k-th composition in standard order is not biquanimous.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 40, 42, 48, 49, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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 terms and corresponding compositions begin:
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  12: (1,3)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)

The unordered complement is A357976, counted by A002219.
The unordered version is A371731, counted by A371795, even case A006827.
These compositions are counted by A371956.
The complement is A372120, counted by A064914.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]

A372120 Numbers k such that the k-th composition in standard order is biquanimous.

Original entry on oeis.org

0, 3, 10, 11, 13, 14, 15, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 136, 137, 138, 139, 140, 141, 142, 143, 145, 147, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 162, 163, 165, 166, 167, 168, 169

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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 terms and corresponding compositions begin:
   0: ()
   3: (1,1)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  36: (3,3)
  37: (3,2,1)
  38: (3,1,2)
  39: (3,1,1,1)
  41: (2,3,1)
  43: (2,2,1,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  50: (1,3,2)
  51: (1,3,1,1)
  52: (1,2,3)
  53: (1,2,2,1)
  54: (1,2,1,2)

These compositions are counted by A064914.
The unordered version (integer partitions) is A357976, counted by A002219.
The unordered complement is A371731, counted by A371795, even case A006827.
The complement is A372119, counted by A371956.
A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
Cf. A027187, A035470, A357879, A367094, A371781, A371782, A371783.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[Total/@Subsets[stc[#]], Total[stc[#]]/2]&]

A371954 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 5, 3, 0, 1, 0, 7, 0, 0, 0, 1, 0, 11, 6, 4, 0, 0, 1, 0, 15, 0, 0, 0, 0, 0, 1, 0, 22, 14, 0, 5, 0, 0, 0, 1, 0, 30, 0, 10, 0, 0, 0, 0, 0, 1, 0, 42, 25, 0, 0, 6, 0, 0, 0, 0, 1, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 77, 53, 30, 15, 0, 7, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 20 2024

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.

			Triangle begins:
  1
  0  1
  0  2  1
  0  3  0  1
  0  5  3  0  1
  0  7  0  0  0  1
  0 11  6  4  0  0  1
  0 15  0  0  0  0  0  1
  0 22 14  0  5  0  0  0  1
  0 30  0 10  0  0  0  0  0  1
  0 42 25  0  0  6  0  0  0  0  1
  0 56  0  0  0  0  0  0  0  0  0  1
  0 77 53 30 15  0  7  0  0  0  0  0  1
Row n = 6 counts the following partitions:
  .  (6)       (33)      (222)     .  .  (111111)
     (51)      (321)     (2211)
     (42)      (3111)    (21111)
     (411)     (2211)    (111111)
     (33)      (21111)
     (321)     (111111)
     (3111)
     (222)
     (2211)
     (21111)
     (111111)

Row n has A000005(n) positive entries.
Column k = 1 is A000041.
Column k = 2 is A002219 (aerated), ranks A357976.
Column k = 3 is A002220 (aerated), ranks A371955.
Removing all zeros gives A371783.
Row sums are A372121.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371789 counts non-quanimous sets, complement A371796.
Cf. A002221, A002222, A006827, A035470, A064914, A237258, A321142, A321455, A365543, A371795.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,0,10},{k,0,n}]

A372122 Number of strict triquanimous partitions of 3n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 5, 13, 18, 36, 51, 93, 132, 229, 315, 516, 735, 1134, 1575, 2407, 3309, 4878, 6710, 9690, 13168, 18744, 25114, 35050, 47210, 64503, 85573, 116445, 153328, 205367, 269383, 356668, 464268, 610644, 788274, 1026330, 1321017, 1704309, 2176054

Offset: 0

Gus Wiseman, Apr 20 2024

nonn

A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums. Triquanimous partitions are counted by A002220 and ranked by A371955.

			The partition (11,7,5,4,3,2,1) has qualifying set partitions {{11},{4,7},{1,2,3,5}} and {{11},{1,3,7},{2,4,5}} so is counted under a(11).
The a(5) = 1 through a(9) = 13 partitions:
  (5,4,3,2,1)  (6,5,4,2,1)  (7,5,4,3,2)    (8,6,5,3,2)    (9,6,5,4,3)
                            (7,6,4,3,1)    (8,7,5,3,1)    (9,7,5,4,2)
                            (7,6,5,2,1)    (8,7,6,2,1)    (9,7,6,3,2)
                            (6,5,4,3,2,1)  (7,6,5,3,2,1)  (9,8,5,4,1)
                                           (8,6,4,3,2,1)  (9,8,6,3,1)
                                                          (9,8,7,2,1)
                                                          (7,6,5,4,3,2)
                                                          (8,6,5,4,3,1)
                                                          (8,7,5,4,2,1)
                                                          (8,7,6,3,2,1)
                                                          (9,6,5,4,2,1)
                                                          (9,7,5,3,2,1)
                                                          (9,8,4,3,2,1)

The non-strict biquanimous version is A002219, ranks A357976.
The non-strict version is A002220, ranks A371955.
The biquanimous version is A237258, ranks A357854.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454, strict A371737.
A371783 counts k-quanimous partitions.
A371795 counts non-biquanimous partitions, even case A006827, ranks A371731.
Cf. A035470, A064914, A321142, A321455, A371781, A371796.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[3n], UnsameQ@@#&&Select[facs[Times@@Prime/@#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]],{n,0,10}]

More terms from Jinyuan Wang, Mar 30 2025

cp's OEIS Frontend

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

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

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A371839 Number of integer partitions of n with biquanimous multiplicities.

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

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A371955 Numbers with triquanimous prime indices.

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A371956 Number of non-biquanimous compositions of 2n.

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A372119 Numbers k such that the k-th composition in standard order is not biquanimous.

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A372120 Numbers k such that the k-th composition in standard order is biquanimous.

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A371954 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).

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