A371783 -id:A371783 - cp's OEIS frontend

A371789 Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.

1, 2, 4, 7, 13, 24, 45, 85, 162, 306, 585, 1102, 2106, 3988, 7623, 14535, 27758, 52921, 101848, 195618, 378383, 733609, 1421868, 2755807, 5373060, 10482925, 20495335, 40119622, 78476107, 153463714, 300732073

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 not counted under a(9).
The a(0) = 1 through a(4) = 13 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}

The "bi-" complement for integer partitions is A002219, ranks A357976.
The "bi-" complement for strict partitions is A237258, ranks A357854.
The version for integer partitions is A321451, ranks A321453.
The complement for integer partitions is A321452, ranks A321454
The version for strict partitions is A371736, complement A371737.
First differences are A371790.
The "bi-" version is A371792, complement A371791.
The "bi-" version for strict partitions is A371794 (bisection A321142).
The "bi-" version for integer partitions is A371795, ranks A371731.
The complement is counted by A371796, differences 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, A365663, A365661.

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,0,8}]

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

A371737 Number of quanimous strict integer partitions of n, meaning there is more than one set partition with all equal block-sums.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 4, 0, 7, 1, 9, 0, 16, 0, 21, 4, 32, 0, 45, 0, 63, 13, 84, 0, 126, 0, 158, 36, 220, 0, 303, 0, 393, 93, 511, 0, 708, 0, 881, 229, 1156, 0, 1539, 0, 1925, 516, 2445, 0, 3233, 6, 3952, 1134, 5019, 0, 6497

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.

Conjecture: (1) Positions of 0's are A327782. (2) Positions of terms > 0 are A368459.

			The a(0) = 0 through a(14) = 7 strict partitions:
  .  .  .  .  .  .  (321)  .  (431)  .  (532)   .  (642)   .  (743)
                                        (541)      (651)      (752)
                                        (4321)     (5421)     (761)
                                                   (6321)     (5432)
                                                              (6431)
                                                              (6521)
                                                              (7421)

The non-strict "bi-" version is A002219, ranks A357976.
The "bi-" version is A237258, ranks A357854, complement A321142 or A371794.
The non-strict version is A321452, ranks A321454.
The complement is A371736, non-strict A321451, ranks A321453.
The non-strict "bi-" complement is A371795, ranks A371731.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous sets, complement A371792.
A371796 counts quanimous sets, complement A371789.
Cf. A000005, A018818, A035470, A038041, A064688, A232466, A237194, A305551, A365663, A365661, A365925, A371733, A371839.

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}]

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 *)

A371797 Number of quanimous subsets of {1..n} containing n, meaning there is more than one set partition with equal block-sums.

Original entry on oeis.org

0, 0, 1, 2, 5, 11, 24, 51, 112, 233, 507, 1044, 2214, 4557, 9472, 19545, 40373, 82145, 168374, 341523, 693350, 1408893, 2860365, 5771355, 11667351, 23542022, 47484577, 95861243, 193447849, 389602553

Offset: 1

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(1) = 0 through a(6) = 11 subsets:
  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                 {1,2,3,4}  {2,3,5}      {2,4,6}
                            {1,2,4,5}    {1,2,3,6}
                            {2,3,4,5}    {1,2,5,6}
                            {1,2,3,4,5}  {1,3,4,6}
                                         {2,3,5,6}
                                         {3,4,5,6}
                                         {1,2,3,4,6}
                                         {1,2,4,5,6}
                                         {2,3,4,5,6}
                                         {1,2,3,4,5,6}

The "bi-" version is A232466, complement A371793.
The complement is counted by A371790.
First differences of A371796, complement A371789.
A371736 counts non-quanimous strict partitions.
A371737 counts quanimous strict partitions.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous subsets, complement A371792.
Cf. A000005, A002219, A035470, A038041, A275972, A279791, A321452, A371795.

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

a(11)-a(30) from Martin Fuller, Apr 01 2025

A371782 Numbers with non-biquanimous prime signature.

Original entry on oeis.org

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 *)

A371790 Number of non-quanimous subsets of {1..n} containing n, meaning there is only one set partition with equal block-sums.

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 40, 77, 144, 279, 517, 1004, 1882, 3635, 6912, 13223, 25163, 48927, 93770, 182765, 355226, 688259, 1333939, 2617253, 5109865, 10012410, 19624287, 38356485, 74987607, 147268359

Offset: 1

Gus Wiseman, Apr 17 2024

			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 not counted under a(9).
The a(1) = 1 through a(5) = 11 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}

First differences of A371789, complement counted by A371796.
The "bi-" version is A371793, complement A232466.
The complement is counted by A371797.
A371736 counts non-quanimous strict partitions.
A371737 counts quanimous strict partitions.
A371783 counts k-quanimous partitions.
A371791 counts biquanimous subsets, complement A371792.
Cf. A002219, A035470, A038041, A275972, A316402, A321451, A321452.

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

a(11)-a(30) from Martin Fuller, Apr 01 2025

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

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

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

cp's OEIS Frontend

A371789 Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.

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A371737 Number of quanimous strict integer partitions of n, meaning there is more than one set partition with all equal block-sums.

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A371781 Numbers with biquanimous prime signature.

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A371797 Number of quanimous subsets of {1..n} containing n, meaning there is more than one set partition with equal block-sums.

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A371782 Numbers with non-biquanimous prime signature.

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A371790 Number of non-quanimous subsets of {1..n} containing n, meaning there is only one set partition with equal block-sums.

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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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