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

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

Author

Gus Wiseman, Mar 14 2022

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

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]&]

Formula

Intersection of A350944 and A350945.

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

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

Closed under A122111 (conjugation).