A371839 -id:A371839 - cp's OEIS frontend

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

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 *)

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 *)

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

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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A371782 Numbers with non-biquanimous prime signature.

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

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