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A350945
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Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.
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23
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1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 36, 38, 39, 41, 44, 47, 56, 57, 58, 59, 66, 67, 68, 73, 74, 75, 80, 83, 84, 86, 87, 92, 96, 97, 102, 103, 104, 106, 109, 111, 120, 122, 124, 125, 127, 128, 129, 137, 138, 142, 144, 149, 152, 156
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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: ()
2: (1)
5: (3)
6: (2,1)
8: (1,1,1)
9: (2,2)
11: (5)
14: (4,1)
17: (7)
20: (3,1,1)
21: (4,2)
23: (9)
24: (2,1,1,1)
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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[primeMS[#], _?EvenQ]&]
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CROSSREFS
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These partitions are counted by A350948.
These are the positions of 0's in A350950.
A122111 = conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
The following rank partitions:
A349157: # of even parts = # of odd conjugate parts, counted by A277579.
A350848: # of even conj parts = # of odd conj parts, counted by A045931.
A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A000070, A000290, A027187, A027193, A103919, A236559, A344607, A344651, A345196, `A350942, A350950, A350951.
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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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