A257992 -id:A257992 - cp's OEIS frontend

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

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)).

A350949 Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.

Original entry on oeis.org

1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 294, 315, 350, 416, 441, 490, 525, 624, 660, 735, 924, 990, 1088, 1100, 1386, 1540, 1560, 1632, 1650, 1715, 2184, 2310, 2340, 2401, 2432, 2600, 3267, 3276, 3388, 3640, 3648, 3900, 4080, 4125, 5082, 5324, 5390

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: ()
     2: (1)
     6: (2,1)
     9: (2,2)
    20: (3,1,1)
    30: (3,2,1)
    56: (4,1,1,1)
    75: (3,3,2)
    84: (4,2,1,1)
   125: (3,3,3)
   176: (5,1,1,1,1)
   210: (4,3,2,1)
   264: (5,2,1,1,1)
   294: (4,4,2,1)
   315: (4,3,2,2)
   350: (4,3,3,1)
   416: (6,1,1,1,1,1)

The second condition alone is A350944, counted by A277103.
The first condition alone is A350945, counted by A350948.
The case of all four statistics equal is A350947, counted by A351978.
These partitions are counted by A351976.
There are four other possible pairings of statistics:
- A045931: # even = # odd, ranked by A325698, strict A239241.
- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
- A277579: # even = # odd conj, ranked by A349157, strict A352131.
- A277579: # even conj = # odd, ranked by A350943, strict A352130.
There are two other possible double-pairings of statistics:
- A350946, counted by A351977.
- 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.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A130780, A171966, 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[#],?OddQ]==Count[conj[primeMS[#]],?OddQ]&&Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?EvenQ]&]

Intersection of A350944 and A350945.
A257991(a(n)) = A344616(a(n)).
A257992(a(n)) = A350847(a(n)).
Closed under A122111 (conjugation).

A324929 Numbers whose product of prime indices is even.

Original entry on oeis.org

3, 6, 7, 9, 12, 13, 14, 15, 18, 19, 21, 24, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 42, 43, 45, 48, 49, 51, 52, 53, 54, 56, 57, 58, 60, 61, 63, 65, 66, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 81, 84, 86, 87, 89, 90, 91, 93, 95, 96, 98, 99, 101, 102, 104

Offset: 1

Gus Wiseman, Mar 21 2019

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, with product A003963(n).

Also Heinz numbers of integer partitions whose product of parts is even (counted by A047967), where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   3: {2}
   6: {1,2}
   7: {4}
   9: {2,2}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  36: {1,1,2,2}

Cf. A000720, A001222, A003963, A005087, A033844, A047967, A066208, A112798, A257992, A318400, A324927.

Select[Range[100],EvenQ[Times@@PrimePi/@If[#==1,{},FactorInteger[#]][[All,1]]]&]

isok(n) = my(f=factor(n)[,1]); !(prod(k=1, #f, primepi(f[k])) % 2); \\ Michel Marcus, Mar 22 2019

A379301 Positive integers whose prime indices include a unique composite number.

Original entry on oeis.org

7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131

Offset: 1

Gus Wiseman, Dec 25 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.

			The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.

For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of one in A379300.
Partitions of this type are counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A113646, A376680.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],CompositeQ]]==1&]

A350849 Number of odd conjugate parts minus number of even parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 3, 0, 3, 1, -2, 2, 5, 1, 5, 2, 0, 0, 7, -1, 7, 3, 0, 4, 9, 0, 0, 4, -1, 3, 9, 1, 11, 1, 2, 6, 0, -2, 11, 6, 2, 2, 13, 1, 13, 5, 1, 8, 15, 1, -2, 1, 4, 5, 15, -2, 2, 2, 4, 8, 17, 0, 17, 10, 1, 0, 2, 3, 19, 7, 6, 1, 19, -1, 21, 10, 1, 7, 0, 3, 21, 3

Offset: 1

Gus Wiseman, Jan 28 2022

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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) = 4, 3, 2, 1, 0, -1, -2, -3, -4, together with their prime indices, are:
   22: (5,1)
    5: (3)
   10: (3,1)
    2: (1)
    1: ()
   18: (2,2,1)
    9: (2,2)
  162: (2,2,2,2,1)
   81: (2,2,2,2)

This is a hybrid of A195017 and A350941.
Positions of 0's are A349157.
Counting even conjugate parts instead of even parts gives A350941.
The conjugate version is A350942.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
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.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
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[conj[primeMS[n]],?OddQ]-Count[primeMS[n],?EvenQ],{n,100}]

a(n) = A344616(n) - A257992(n).

cp's OEIS Frontend

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

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A350949 Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.

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A324929 Numbers whose product of prime indices is even.

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PARI

A379301 Positive integers whose prime indices include a unique composite number.

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