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A303280
Number of strict integer partitions of n whose parts have a common divisor other than 1.
10
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
OFFSET
1,6
LINKS
FORMULA
a(n) = -Sum_{d|n, d > 1} mu(d) * A000009(n/d).
EXAMPLE
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).
MAPLE
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}):
seq(a(n), n=1..100); # Alois P. Heinz, Apr 23 2018
MATHEMATICA
Table[-Sum[MoebiusMu[d]*PartitionsQ[n/d], {d, Rest[Divisors[n]]}], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 20 2018
STATUS
approved