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A350944
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Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.
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22
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1, 2, 6, 9, 10, 12, 15, 18, 20, 30, 35, 49, 54, 55, 56, 70, 75, 77, 81, 84, 88, 90, 98, 108, 110, 112, 125, 132, 135, 143, 154, 162, 168, 169, 176, 180, 187, 210, 221, 260, 264, 270, 286, 294, 315, 323, 330, 338, 340, 350, 361, 363, 364, 374, 391, 416, 420
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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)
6: (2,1)
9: (2,2)
10: (3,1)
12: (2,1,1)
15: (3,2)
18: (2,2,1)
20: (3,1,1)
30: (3,2,1)
35: (4,3)
49: (4,4)
54: (2,2,2,1)
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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[#]], _?OddQ]==Count[primeMS[#], _?OddQ]&]
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CROSSREFS
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These partitions are counted by A277103.
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.
A350945: # of even parts = # of even conjugate parts, counted by A350948.
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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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