A257991 -id:A257991 - cp's OEIS frontend

A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

1, 4, 6, 15, 16, 21, 24, 25, 35, 60, 64, 77, 84, 90, 91, 96, 100, 121, 126, 140, 143, 150, 210, 221, 240, 247, 256, 289, 297, 308, 323, 336, 351, 360, 364, 375, 384, 400, 437, 462, 484, 490, 495, 504, 525, 529, 546, 551, 560, 572, 585, 600, 625, 667, 686, 726

Offset: 1

Gus Wiseman, Jan 21 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with the same number of even prime indices as odd conjugate prime indices.

These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.

			The terms and their prime indices begin:
    1: ()
    4: (1,1)
    6: (2,1)
   15: (3,2)
   16: (1,1,1,1)
   21: (4,2)
   24: (2,1,1,1)
   25: (3,3)
   35: (4,3)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
   77: (5,4)
   84: (4,2,1,1)
   90: (3,2,2,1)
   91: (6,4)
   96: (2,1,1,1,1,1)

A subset of A028260 (even bigomega), counted by A027187.
These partitions are counted by A277579.
This is the half-conjugate version of A325698, counted by A045931.
A000041 counts partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A100824 counts partitions with at most one odd part, ranked by A349150.
A108950/A108949 count partitions with more odd/even parts.
A122111 represents conjugation using Heinz numbers.
A130780/A171966 count partitions with more or equal odd/even parts.
A257991/A257992 count odd/even prime indices.
A316524 gives the alternating sum of prime indices (reverse: A344616).
Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(A122111(a(n))).

A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 2, 1, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 4, 2, 2, 0, 0, 1, 3, 1, 2, 1, 0, 2, 0, 1, 0, 1, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 1, 0, 4, 1, 0, 0, 2, 1, 0, 2, 3, 1, 2

Offset: 1

Gus Wiseman, Mar 14 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) counts even prime indices of n.

Positions of first appearances are A001248.
The triangular version is A116482.
Positions of zeros are A346635.
Subtracting from the number of odd conjugate parts gives A350941.
Subtracting from the number of odd parts gives A350942.
Subtracting from the number of even parts gives A350950.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847 (this sequence).
There are six possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
Cf. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Count[conj[primeMS[n]],_?EvenQ],{n,100}]

a(n) = A344616(n) - A350941(n).

a(n) = A257992(A122111(n)).

A350848 Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 6, 18, 21, 24, 54, 65, 70, 72, 84, 96, 133, 147, 162, 182, 189, 210, 216, 260, 280, 288, 319, 336, 384, 418, 429, 481, 486, 490, 525, 532, 546, 585, 588, 630, 648, 728, 731, 741, 754, 756, 840, 845, 864, 1007, 1029, 1040, 1120, 1152, 1197, 1254, 1258, 1276

Offset: 1

Gus Wiseman, Jan 27 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
   1: ()
   6: (2,1)
  18: (2,2,1)
  21: (4,2)
  24: (2,1,1,1)
  54: (2,2,2,1)
  65: (6,3)
  70: (4,3,1)
  72: (2,2,1,1,1)
  84: (4,2,1,1)
  96: (2,1,1,1,1,1)

These partitions are counted by A045931.
The conjugate strict version is counted by A239241.
The conjugate version is A325698.
These are the positions of 0's in A350941.
Adding the conjugate condition gives A350946, all four equal A350947.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A325698: # of even parts = # of odd parts.
A349157: # of even parts = # of odd conjugate parts, counted by A277579.
A350848: # even conjugate parts = # odd conjugate parts, counted by A045931.
A350943: # of even conjugate parts = # of odd parts, counted by A277579.
A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
A350945: # of even parts = # of even conjugate parts, counted by A350948.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A316524 = alternating sum of prime indices, reverse A344616.
Cf. A024619, A026424, A028260, A103919, A130780, A171966, A195017, A241638, A325700, A350849, A350942, A350949.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[conj[primeMS[#]],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A344616(a(n)) = A350847(a(n)).

A257991(A122111(a(n))) = A257992(A122111(a(n))).

A350945 Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 36, 38, 39, 41, 44, 47, 56, 57, 58, 59, 66, 67, 68, 73, 74, 75, 80, 83, 84, 86, 87, 92, 96, 97, 102, 103, 104, 106, 109, 111, 120, 122, 124, 125, 127, 128, 129, 137, 138, 142, 144, 149, 152, 156

Offset: 1

Gus Wiseman, Jan 28 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
   1: ()
   2: (1)
   5: (3)
   6: (2,1)
   8: (1,1,1)
   9: (2,2)
  11: (5)
  14: (4,1)
  17: (7)
  20: (3,1,1)
  21: (4,2)
  23: (9)
  24: (2,1,1,1)

These partitions are counted by A350948.
These are the positions of 0's in A350950.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 = conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325040: product = product of conjugate, counted by A325039.
  A325698: # of even parts = # of odd parts, counted by A045931.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # of even conj parts = # of odd conj parts, counted by A045931.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, `A350942, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[conj[primeMS[#]],?EvenQ]==Count[primeMS[#],?EvenQ]&]

A257992(a(n)) = A350847(a(n)).

A350944 Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 2, 6, 9, 10, 12, 15, 18, 20, 30, 35, 49, 54, 55, 56, 70, 75, 77, 81, 84, 88, 90, 98, 108, 110, 112, 125, 132, 135, 143, 154, 162, 168, 169, 176, 180, 187, 210, 221, 260, 264, 270, 286, 294, 315, 323, 330, 338, 340, 350, 361, 363, 364, 374, 391, 416, 420

Offset: 1

Gus Wiseman, Jan 28 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
   1: ()
   2: (1)
   6: (2,1)
   9: (2,2)
  10: (3,1)
  12: (2,1,1)
  15: (3,2)
  18: (2,2,1)
  20: (3,1,1)
  30: (3,2,1)
  35: (4,3)
  49: (4,4)
  54: (2,2,2,1)

These partitions are counted by A277103.
The even rank case is A345196.
The conjugate version is A350943, counted by A277579.
These are the positions of 0's in A350951, even A350950.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 = conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325040: product = product of conjugate, counted by A325039.
  A325698: # of even parts = # of odd parts, counted by A045931.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # even conj parts = # odd conj parts, counted by A045931.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000700, A027187, A027193, A103919, A236559, A350942.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[conj[primeMS[#]],?OddQ]==Count[primeMS[#],?OddQ]&]

A257991(a(n)) = A344616(a(n)).

A350948 Number of integer partitions of n with as many even parts as even conjugate parts.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 3, 7, 6, 10, 10, 18, 19, 27, 31, 40, 47, 65, 75, 98, 115, 142, 170, 217, 257, 316, 376, 458, 544, 671, 792, 952, 1129, 1351, 1598, 1919, 2259, 2681, 3155, 3739, 4384, 5181, 6064, 7129, 8331, 9764, 11380, 13308, 15477, 18047, 20944

Offset: 0

Gus Wiseman, Mar 14 2022

nonn

			The a(0) = 1 through a(8) = 6 partitions (empty column indicated by dot):
  ()  (1)  .  (3)    (22)  (5)      (42)    (7)        (62)
              (21)         (41)     (321)   (61)       (332)
              (111)        (311)    (2211)  (511)      (521)
                           (2111)           (4111)     (4211)
                           (11111)          (31111)    (32111)
                                            (211111)   (221111)
                                            (1111111)
For example, both (3,2,1,1,1) and its conjugate (5,2,1) have exactly 1 even part, so are counted under a(8).

Comparing even to odd parts gives A045931, ranked by A325698.
The odd version is A277103, even rank case A345196, ranked by A350944.
Comparing even to odd conjugate parts gives A277579, ranked by A349157.
Comparing product of parts to product of conjugate parts gives A325039.
These partitions are ranked by A350945, the zeros of A350950.
A000041 counts integer partitions, strict A000009.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A116482 counts partitions by number of even (or even conjugate) parts.
A122111 represents partition conjugation using Heinz numbers.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A351976: # even = # even conj, # odd = # odd conj, ranked by A350949.
A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.
A351978: # even = # odd = # even conj = # odd conj, ranked by A350947.
A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.
Cf. A027187, A130780, A171966, A195017, A239241, A241638, A344607, A344651, A350848, A350941, A350942, A350943.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Count[#,?EvenQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

Original entry on oeis.org

2, 5, 6, 11, 14, 17, 18, 23, 26, 31, 35, 38, 41, 42, 47, 54, 58, 59, 65, 67, 73, 74, 78, 83, 86, 95, 97, 98, 103, 106, 109, 114, 122, 126, 127, 137, 142, 143, 145, 149, 157, 158, 162, 167, 174, 178, 179, 182, 185, 191, 197, 202, 209, 211, 214, 215, 222, 226

Offset: 1

Gus Wiseman, Feb 12 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only odd prime index (counting multiplicity) is the smallest.

			The sequence of partitions together with their Heinz numbers begins:
      2: (1)         54: (2,2,2,1)    109: (29)
      5: (3)         58: (10,1)       114: (8,2,1)
      6: (2,1)       59: (17)         122: (18,1)
     11: (5)         65: (6,3)        126: (4,2,2,1)
     14: (4,1)       67: (19)         127: (31)
     17: (7)         73: (21)         137: (33)
     18: (2,2,1)     74: (12,1)       142: (20,1)
     23: (9)         78: (6,2,1)      143: (6,5)
     26: (6,1)       83: (23)         145: (10,3)
     31: (11)        86: (14,1)       149: (35)
     35: (4,3)       95: (8,3)        157: (37)
     38: (8,1)       97: (25)         158: (22,1)
     41: (13)        98: (4,4,1)      162: (2,2,2,2,1)
     42: (4,2,1)    103: (27)         167: (39)
     47: (15)       106: (16,1)       174: (10,2,1)

These partitions are counted by A035363 (shifted left once).
Terms of A340932 can be factored into elements of this sequence.
The even version is A341447.
A001222 counts prime factors.
A005408 lists odd numbers.
A026804 counts partitions whose smallest part is odd.
A027193 counts odd-length partitions, ranked by A026424.
A031368 lists odd-indexed primes.
A032742 selects largest proper divisor.
A055396 selects smallest prime index.
A056239 adds up prime indices.
A058695 counts partitions of odd numbers, ranked by A300063.
A061395 selects largest prime index.
A066207 lists numbers with all even prime indices.
A066208 lists numbers with all odd prime indices.
A112798 lists the prime indices of each positive integer.
A244991 lists numbers whose greatest prime index is odd.
A340932 lists numbers whose smallest prime index is odd.
Cf. A006141, A160786, A257991, A257992, A300272, A340604, A340607, A340854/A340855, A340933.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[First[primeMS[#]]]&&And@@EvenQ[Rest[primeMS[#]]]&]

Also numbers n > 1 such that A055396(n) is odd and A032742(n) belongs to A066207.

A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

Original entry on oeis.org

1, 3, 6, 7, 13, 14, 18, 19, 26, 27, 29, 36, 37, 38, 42, 43, 53, 54, 58, 61, 63, 70, 71, 74, 78, 79, 84, 86, 89, 101, 105, 106, 107, 113, 114, 117, 122, 126, 130, 131, 139, 140, 142, 151, 156, 158, 162, 163, 171, 173, 174, 178, 181, 190, 193, 195, 199, 202, 210

Offset: 1

Gus Wiseman, Jan 28 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
   1: ()
   3: (2)
   6: (2,1)
   7: (4)
  13: (6)
  14: (4,1)
  18: (2,2,1)
  19: (8)
  26: (6,1)
  27: (2,2,2)
  29: (10)
  36: (2,2,1,1)
  37: (12)
  38: (8,1)
  42: (4,2,1)
For example, the partition (6,3,2) has conjugate (3,3,2,1,1,1) and 1 = 1 so 195 is in the sequence.

These partitions are counted by A277579.
The conjugate version is A349157, also counted by A277579.
These are the positions of 0's in A350942.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 = conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325040: product = product of conjugate, counted by A325039.
  A325698: # of even parts = # of odd parts, counted by A045931.
  A350848: # of even conj parts = # of odd conj parts, counted by A045931.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]&]

A350847(a(n)) = A257991(a(n)).

A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 0, 3, -2, 1, 1, 2, 0, 0, -1, 3, 1, 0, 0, 3, -2, 1, 1, 2, -1, 0, 0, 2, 0, 1, 1, 5, -1, 1, -2, 0, 0, 0, -2, 3, 1, 0, 0, 3, 1, 1, 1, 4, -4, 1, -1, 2, 0, 0, -1, 2, -2, 0, 1, 1, 0, 1, 0, 5, -2, 1, 1, 3, -1, 0, 0, 2, 1, 0, 1, 2, -3, 0, 0, 5, -2, 1

Offset: 1

Gus Wiseman, Jan 28 2022

sign

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			First positions n such that a(n) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are:
  192: (2,1,1,1,1,1,1)
   32: (1,1,1,1,1)
   48: (2,1,1,1,1)
    8: (1,1,1)
   12: (2,1,1)
    2: (1)
    1: ()
   15: (3,2)
    9: (2,2)
   77: (5,4)
   49: (4,4)
  221: (7,6)
  169: (6,6)

The conjugate version is A350849.
This is a hybrid of A195017 and A350941.
Positions of 0's are A350943.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325698: # of even parts = # of odd parts.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # even conj parts = # odd conj parts, counted by A045931.
  A350943: # of even conjugate parts = # of odd parts, counted by A277579.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Count[primeMS[n],?OddQ]-Count[conj[primeMS[n]],?EvenQ],{n,100}]

A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.

Original entry on oeis.org

1, 6, 84, 210, 490, 525, 2184, 2340, 5460, 9464, 12012, 12740, 12870, 13650, 14625, 19152, 22308, 30030, 34125, 43940, 45144, 55770, 59150, 66066, 70070, 70785, 75075, 79625, 82992, 88920

Offset: 1

Gus Wiseman, Mar 14 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      1: ()
      6: (2,1)
     84: (4,2,1,1)
    210: (4,3,2,1)
    490: (4,4,3,1)
    525: (4,3,3,2)
   2184: (6,4,2,1,1,1)
   2340: (6,3,2,2,1,1)
   5460: (6,4,3,2,1,1)
   9464: (6,6,4,1,1,1)
  12012: (6,5,4,2,1,1)
  12740: (6,4,4,3,1,1)
  12870: (6,5,3,2,2,1)
  13650: (6,4,3,3,2,1)
  14625: (6,3,3,3,2,2)
  19152: (8,4,2,2,1,1,1,1)
For example, the partition (6,6,4,1,1,1) has conjugate (6,3,3,3,2,2), and all four statistics are equal to 3, so 9464 is in the sequence.

These partitions are counted by A351978.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are six possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A239241, A241638, A325700, A350849, A350941, A350942, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[1000],Count[primeMS[#],?EvenQ]==Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(a(n)) = A350847(a(n)) = A344616(a(n)).

cp's OEIS Frontend

A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

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A350847 Number of even parts in the conjugate of the integer partition with Heinz number n.

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A350848 Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.

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A350945 Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.

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A350944 Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.

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A350948 Number of integer partitions of n with as many even parts as even conjugate parts.

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A341446 Heinz numbers of integer partitions whose only odd part is the smallest.

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A350943 Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.

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A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

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