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A343382
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Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.
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17
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1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 9, 13, 18, 21, 26, 34, 38, 48, 57, 67, 81, 99, 110, 133, 157, 183, 211, 250, 282, 330, 380, 437, 502, 575, 648, 748, 852, 967, 1095, 1250, 1405, 1597, 1801, 2029, 2287, 2579, 2883, 3245, 3638, 4077, 4557, 5107, 5691, 6356
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OFFSET
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0,8
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COMMENTS
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Alternative name: Number of strict integer partitions of n that are either (1) empty, or (2) have smallest part not dividing all the others, or (3) have greatest part not 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(11) = 9 partitions (empty columns indicated by dots):
() . . . . (3,2) (3,2,1) (4,3) (5,3) (5,4) (6,4) (6,5)
(5,2) (4,3,1) (7,2) (7,3) (7,4)
(5,2,1) (4,3,2) (5,3,2) (8,3)
(5,3,1) (5,4,1) (9,2)
(7,2,1) (5,4,2)
(4,3,2,1) (6,3,2)
(6,4,1)
(7,3,1)
(5,3,2,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)||UnsameQ@@#&&!And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]
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CROSSREFS
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The first condition alone gives A341450.
The second condition alone gives A343377.
The version for "and" instead of "or" is A343379.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343337, A343338, A343341, A343342.
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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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