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A350942
Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.
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
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.
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.
Sequence in context: A140736 A284993 A292068 * A140056 A240239 A247044
KEYWORD
sign
AUTHOR
Gus Wiseman, Jan 28 2022
STATUS
approved