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A343346
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Number of integer partitions of n that are empty, have smallest part not dividing all the others, or greatest part not divisible by all the others.
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14
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1, 0, 0, 0, 0, 1, 1, 4, 6, 11, 16, 29, 36, 59, 80, 112, 150, 214, 271, 374, 476, 624, 800, 1045, 1298, 1669, 2088, 2628, 3258, 4087, 5000, 6219, 7602, 9331, 11368, 13877, 16754, 20368, 24536, 29580, 35468, 42624, 50845, 60827, 72357, 86078, 102100, 121101
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OFFSET
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0,8
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COMMENTS
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Alternative name: Number of integer partitions of n with either no part dividing, or no part divisible by all the others.
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LINKS
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EXAMPLE
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The a(0) = 1 through a(10) = 16 partitions (empty columns indicated by dots):
() . . . . (32) (321) (43) (53) (54) (64)
(52) (332) (72) (73)
(322) (431) (432) (433)
(3211) (521) (522) (532)
(3221) (531) (541)
(32111) (3222) (721)
(3321) (3322)
(4311) (4321)
(5211) (5221)
(32211) (5311)
(321111) (32221)
(33211)
(43111)
(52111)
(322111)
(3211111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], #=={}||!And@@IntegerQ/@(#/Min@@#)||!And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]
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CROSSREFS
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The complement is counted by A130714.
The first condition alone gives A338470.
The second condition alone gives A343341.
The "and" instead of "or" version is A343342.
The Heinz numbers of these partitions are A343343.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
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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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