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A303280
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Number of strict integer partitions of n whose parts have a common divisor other than 1.
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10
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0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 1, 10, 1, 11, 6, 12, 1, 19, 3, 18, 8, 23, 1, 36, 1, 32, 13, 38, 7, 57, 1, 54, 19, 68, 1, 95, 1, 90, 33, 104, 1, 148, 5, 149, 39, 166, 1, 230, 14, 226, 55, 256, 1, 360, 1, 340, 82, 390, 20, 527, 1, 513, 105, 609, 1
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OFFSET
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1,6
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LINKS
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FORMULA
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a(n) = -Sum_{d|n, d > 1} mu(d) * A000009(n/d).
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EXAMPLE
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The a(18) = 10 strict partitions are (18), (10,8), (12,6), (14,4), (15,3), (16,2), (8,6,4), (9,6,3), (10,6,2), (12,4,2).
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MAPLE
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with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
`if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
end:
a:= n-> -add(mobius(d)*b(n/d), d=divisors(n) minus {1}):
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MATHEMATICA
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Table[-Sum[MoebiusMu[d]*PartitionsQ[n/d], {d, Rest[Divisors[n]]}], {n, 100}]
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CROSSREFS
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Cf. A000009, A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303138.
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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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