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A350941
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Number of odd conjugate parts minus number of even conjugate parts in the integer partition with Heinz number n.
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15
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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)
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MATHEMATICA
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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}]
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CROSSREFS
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A122111 represents conjugation using Heinz numbers.
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.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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