A371797 -id:A371797 - cp's OEIS frontend

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

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

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

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}]

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&]

A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

Original entry on oeis.org

1, 3, 4, 9, 8, 22, 16, 42, 41, 74, 57, 183, 102, 233, 263, 463, 298, 875, 491, 1350, 1172, 1775, 1256, 4273, 2225, 4399, 4584, 8049, 4566, 14913, 6843, 18539, 15831, 22894, 18196, 53323, 21638, 48947, 50281, 94500, 44584, 144976, 63262, 173436, 169361, 202153

Offset: 1

Gus Wiseman, Apr 20 2024

nonn

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums. The triangles A371783 and A371954 count k-quanimous partitions.

Row sums of A371783.
Row sums of A371954.
A000005 counts divisors.
A000041 counts integer partitions.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A321452 counts quanimous partitions, complement A321451.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A035470, A064914, A321455, A365543, A371737, A371791, 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[Sum[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{k,Divisors[n]}],{n,1,10}]

T(n, d) = my(v=partitions(n/d), w=List([])); forvec(s=vector(d, i, [1, #v]), listput(w, vecsort(concat(vector(d, i, v[s[i]])))), 1); #Set(w);
a(n) = sumdiv(n, d, T(n, d)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

cp's OEIS Frontend

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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A371733 Maximal length of a factorization of n into factors > 1 all having the same sum 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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A371738 Numbers with non-quanimous binary indices. Numbers whose binary indices have only one set partition with all equal block-sums.

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

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A372121 Row sums of A371783 and A371954 (k-quanimous partitions).

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