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A349156
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Number of integer partitions of n whose mean is not an integer.
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35
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1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752
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OFFSET
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0,6
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COMMENTS
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Equivalently, partitions whose length does not divide their sum.
By conjugation, also the number of integer partitions of n with greatest part not dividing n.
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LINKS
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FORMULA
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EXAMPLE
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The a(3) = 1 through a(8) = 11 partitions:
(21) (211) (32) (2211) (43) (332)
(41) (3111) (52) (422)
(221) (21111) (61) (431)
(311) (322) (521)
(2111) (331) (611)
(421) (22211)
(511) (32111)
(2221) (41111)
(3211) (221111)
(4111) (311111)
(22111) (2111111)
(31111)
(211111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], !IntegerQ[Mean[#]]&]], {n, 0, 30}]
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CROSSREFS
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Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The version for distinct prime factors is A176587, complement A078174.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.
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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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