A350848 -id:A350848

A350847

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

+10
23

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,9

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..87.

FORMULA

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

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

MATHEMATICA

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}]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 14 2022

STATUS

approved

A350945

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

+10
23

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..61.

FORMULA

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

EXAMPLE

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)

MATHEMATICA

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]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 28 2022

STATUS

approved

A350944

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

+10
22

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..57.

FORMULA

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

EXAMPLE

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)

MATHEMATICA

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]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 28 2022

STATUS

approved

A350948

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

+10
22

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

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).

MATHEMATICA

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}]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 14 2022

STATUS

approved

A350943

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

+10
21

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..59.

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 28 2022

STATUS

approved

A350942

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

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..82.

EXAMPLE

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)

MATHEMATICA

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}]

CROSSREFS

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.

KEYWORD

sign

AUTHOR

Gus Wiseman, Jan 28 2022

STATUS

approved

A350947

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

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..30.

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 14 2022

STATUS

approved

A350949

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

+10
19

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..50.

FORMULA

Intersection of A350944 and A350945.

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

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

Closed under A122111 (conjugation).

EXAMPLE

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)

MATHEMATICA

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]&]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 14 2022

STATUS

approved

A350849

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

+10
16

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..80.

FORMULA

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

EXAMPLE

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)

MATHEMATICA

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}]

CROSSREFS

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.

KEYWORD

sign

AUTHOR

Gus Wiseman, Jan 28 2022

STATUS

approved

A350946

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

+10
16

1, 6, 65, 84, 210, 216, 319, 490, 525, 532, 731, 1254, 1403, 1924, 2184, 2340, 2449, 2470, 3024, 3135, 3325, 3774, 4028, 4141, 4522, 5311, 5460, 7030, 7314, 7315, 7560, 7776, 7942, 8201, 8236, 9048, 9435, 9464, 10659, 10921, 11484, 11914, 12012, 12025, 12740

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OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..45.

FORMULA

Closed under A122111 (conjugation).

Intersection of A325698 and A350848.

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

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

EXAMPLE

The terms together with their prime indices begin:

1: ()

6: (2,1)

65: (6,3)

84: (4,2,1,1)

210: (4,3,2,1)

216: (2,2,2,1,1,1)

319: (10,5)

490: (4,4,3,1)

525: (4,3,3,2)

532: (8,4,1,1)

731: (14,7)

1254: (8,5,2,1)

1403: (18,9)

1924: (12,6,1,1)

2184: (6,4,2,1,1,1)

2340: (6,3,2,2,1,1)

2449: (22,11)

2470: (8,6,3,1)

For example, the prime indices of 532 are (8,4,1,1), even/odd counts 2/2, and the prime indices of the conjugate 3024 are (4,2,2,2,1,1,1,1), with even/odd counts 4/4; so 532 belongs to the sequence.

MATHEMATICA

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], #==1||Mean[Mod[primeMS[#], 2]]== Mean[Mod[conj[primeMS[#]], 2]]==1/2&]

CROSSREFS

For the first condition alone:

- counted by A045931 (strict A239241)

- ordered version (compositions) A098123

- ranked by A325698

- without multiplicity A325700 (counted by A241638)

The second condition alone is ranked by A350848, strict A352129.

These partitions are counted by A351977, strict A352128.

There are four statistics:

- A257991 = # of odd parts, conjugate A344616.

- A257992 = # of even parts, conjugate A350847.

There are four other possible pairings of statistics:

- A349157: # of even parts = # of odd conjugate parts, counted by A277579.

- A350943: # of even conj parts = # of odd parts, strict counted by A352130.

- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.

- A350945: # of even parts = # of even conjugate parts, counted by A350948.

There are two other possible double-pairings of statistics:

- 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.

A195017 = # of even parts - # of odd parts.

A316524 = alternating sum of prime indices.

Cf. A026424, A028260, A130780, A171966, A347450, A350849, A350941, A350942, A350950, A350951.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 14 2022

STATUS

approved