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A343378
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Number of strict integer partitions of n that are empty or such that (1) the smallest part divides every other part and (2) the greatest part is divisible by every other part.
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12
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1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 6, 5, 4, 6, 6, 4, 8, 6, 7, 9, 8, 5, 12, 9, 8, 9, 11, 6, 14, 10, 10, 11, 10, 10, 20, 12, 12, 15, 18, 10, 21, 13, 15, 19, 17, 11, 27, 19, 20, 20, 25, 13, 27, 22, 26, 23, 24, 15, 34, 23, 21, 27, 30, 19, 38, 24, 26, 27, 37
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OFFSET
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0,4
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COMMENTS
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Alternative name: Number of strict integer partitions of n with a part dividing all the others and a part divisible by all the others.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
1 2 3 4 5 6 7 8 9 A B C D E F
21 31 41 42 61 62 63 82 A1 84 C1 C2 A5
51 421 71 81 91 821 93 841 D1 C3
621 631 A2 931 842 E1
B1 A21 C21
6321 8421
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&&And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]
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CROSSREFS
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The first condition alone gives A097986.
The non-strict version is A130714 (Heinz numbers are complement of A343343).
The second condition alone gives A343347.
The strict complement is counted by A343382.
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.
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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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