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A350946
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Heinz numbers of integer partitions with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.
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16
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1, 6, 65, 84, 210, 216, 319, 490, 525, 532, 731, 1254, 1403, 1924, 2184, 2340, 2449, 2470, 3024, 3135, 3325, 3774, 4028, 4141, 4522, 5311, 5460, 7030, 7314, 7315, 7560, 7776, 7942, 8201, 8236, 9048, 9435, 9464, 10659, 10921, 11484, 11914, 12012, 12025, 12740
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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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Closed under A122111 (conjugation).
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EXAMPLE
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The terms together with their prime indices begin:
1: ()
6: (2,1)
65: (6,3)
84: (4,2,1,1)
210: (4,3,2,1)
216: (2,2,2,1,1,1)
319: (10,5)
490: (4,4,3,1)
525: (4,3,3,2)
532: (8,4,1,1)
731: (14,7)
1254: (8,5,2,1)
1403: (18,9)
1924: (12,6,1,1)
2184: (6,4,2,1,1,1)
2340: (6,3,2,2,1,1)
2449: (22,11)
2470: (8,6,3,1)
For example, the prime indices of 532 are (8,4,1,1), even/odd counts 2/2, and the prime indices of the conjugate 3024 are (4,2,2,2,1,1,1,1), with even/odd counts 4/4; so 532 belongs to the sequence.
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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[1000], #==1||Mean[Mod[primeMS[#], 2]]== Mean[Mod[conj[primeMS[#]], 2]]==1/2&]
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CROSSREFS
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For the first condition alone:
- ordered version (compositions) A098123
There are four statistics:
There are four other possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350943: # of even conj parts = # of odd parts, strict counted by A352130.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are two other possible double-pairings of statistics:
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
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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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