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A334362
Number of r X s rectangles with integer sides such that r <= s, r + s = 2n and s/r is squarefree.
1
1, 2, 3, 3, 1, 5, 3, 4, 4, 3, 3, 7, 1, 4, 5, 5, 2, 7, 2, 5, 6, 5, 2, 9, 1, 3, 6, 6, 2, 9, 3, 5, 6, 4, 5, 10, 2, 3, 5, 7, 1, 9, 3, 7, 7, 4, 3, 11, 4, 3, 5, 5, 2, 10, 4, 8, 5, 4, 2, 13, 1, 5, 8, 6, 2, 10, 3, 5, 5, 8, 3, 13, 2, 3, 7, 5, 5, 9, 3, 9, 7, 3, 3, 13, 2, 4, 6, 8, 2, 13
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{i=1..n} mu(floor((2*n - i)/i))^2 * (1 - ceiling((2*n - i)/i) + floor((2*n - i)/i)), where mu is the Möbius function (A008683).
EXAMPLE
a(6) = 5; 2*6 = 12 has five rectangles, 1 X 11, 2 X 10, 3 X 9, 4 X 8 and 6 X 6 such that the ratios 11/1 = 11, 10/2 = 5, 9/3 = 3, 8/4 = 2 and 6/6 = 1 are all squarefree.
MAPLE
f:= proc(n) uses numtheory;
nops(select(r -> r <= n and issqrfree(2*n/r-1) , divisors(2*n), n))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 27 2020
MATHEMATICA
Table[Sum[MoebiusMu[(2 n - i)/i]^2 (1 - Ceiling[(2 n - i)/i] + Floor[(2 n - i)/i]), {i, n}], {n, 100}]
CROSSREFS
Cf. A008683.
Sequence in context: A077990 A085667 A220114 * A035516 A120428 A079950
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 24 2020
STATUS
approved