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A350848
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Heinz numbers of integer partitions for which the number of even conjugate parts is equal to the number of odd conjugate parts.
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23
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1, 6, 18, 21, 24, 54, 65, 70, 72, 84, 96, 133, 147, 162, 182, 189, 210, 216, 260, 280, 288, 319, 336, 384, 418, 429, 481, 486, 490, 525, 532, 546, 585, 588, 630, 648, 728, 731, 741, 754, 756, 840, 845, 864, 1007, 1029, 1040, 1120, 1152, 1197, 1254, 1258, 1276
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OFFSET
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1,2
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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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The terms together with their prime indices begin:
1: ()
6: (2,1)
18: (2,2,1)
21: (4,2)
24: (2,1,1,1)
54: (2,2,2,1)
65: (6,3)
70: (4,3,1)
72: (2,2,1,1,1)
84: (4,2,1,1)
96: (2,1,1,1,1,1)
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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]}]];
Select[Range[100], Count[conj[primeMS[#]], _?EvenQ]==Count[conj[primeMS[#]], _?OddQ]&]
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CROSSREFS
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These partitions are counted by A045931.
The conjugate strict version is counted by A239241.
These are the positions of 0's in A350941.
A325698: # of even parts = # of odd parts.
A349157: # of even parts = # of odd conjugate parts, counted by A277579.
A350848: # even conjugate parts = # odd conjugate 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. A024619, A026424, A028260, A103919, A130780, A171966, A195017, A241638, A325700, A350849, A350942, A350949.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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