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A351980
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Heinz numbers of integer partitions with as many even parts as odd conjugate parts and as many odd parts as even conjugate parts.
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15
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1, 6, 84, 126, 140, 210, 490, 525, 686, 875, 1404, 1456, 2106, 2184, 2288, 2340, 3432, 3510, 5460, 6760, 7644, 8190, 8580, 8775, 9100, 9464, 11466, 12012, 12740, 12870, 13650, 14300, 14625, 15808, 18018, 18468, 19110, 19152, 20020, 20672, 21450, 22308, 23712
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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)
84: (4,2,1,1)
126: (4,2,2,1)
140: (4,3,1,1)
210: (4,3,2,1)
490: (4,4,3,1)
525: (4,3,3,2)
686: (4,4,4,1)
875: (4,3,3,3)
1404: (6,2,2,2,1,1)
1456: (6,4,1,1,1,1)
2106: (6,2,2,2,2,1)
2184: (6,4,2,1,1,1)
2288: (6,5,1,1,1,1)
2340: (6,3,2,2,1,1)
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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[1000], Count[primeMS[#], _?EvenQ]==Count[conj[primeMS[#]], _?OddQ]&&Count[primeMS[#], _?OddQ]==Count[conj[primeMS[#]], _?EvenQ]&]
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CROSSREFS
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There are two other possible double-pairings of statistics:
These partitions are counted by A351981.
Partitions with as many even as odd parts:
- strict conjugate case counted by A352129
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
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, A098123, A130780, A171966, A241638, A325700, A350841, A350849, A350941, 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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