A350941 Number of odd conjugate parts minus number of even conjugate parts in the integer partition with Heinz number n.

0, 1, 2, -1, 3, 0, 4, 1, -2, 1, 5, 2, 6, 2, -1, -1, 7, 0, 8, 3, 0, 3, 9, 0, -3, 4, 2, 4, 10, 1, 11, 1, 1, 5, -2, -2, 12, 6, 2, 1, 13, 2, 14, 5, 3, 7, 15, 2, -4, -1, 3, 6, 16, 0, -1, 2, 4, 8, 17, -1, 18, 9, 4, -1, 0, 3, 19, 7, 5, 0, 20, 0, 21, 10, 1, 8, -3, 4

Offset: 0

Author

Gus Wiseman, Jan 28 2022

Keywords

• sign

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

			First positions n such that a(n) = 4, 3, 2, 1, 0, -1, -2, -3, -4, together with their prime indices, are:
   7: (4)
   5: (3)
   3: (2)
   2: (1)
   1: ()
   4: (1,1)
   9: (2,2)
  25: (3,3)
  49: (4,4)
		

Crossrefs

A hybrid with A195017 (non-conjugate version) is A350849, conjugate A350942.

Positions of 0's are A350848, counted by A045931.

A000041 = integer partitions, strict A000009.

A056239 adds up prime indices, counted by A001222, row sums of A112798.

A122111 represents conjugation using Heinz numbers.

A257991 counts odd parts, conjugate A344616.

A257992 counts 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.

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, A350841, A350947, A350949, 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]}]];
Table[Count[conj[primeMS[n]],?OddQ]-Count[conj[primeMS[n]],?EvenQ],{n,1,50}]

Formula

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