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%I #33 Feb 01 2021 19:21:08
%S 1,2,2,2,1,3,2,2,2,3,2,3,1,3,2,2,2,4,2,4,4,4,1,3,0,3,2,2,2,5,2,1,3,4,
%T 3,4,2,3,3,3,1,6,2,4,3,3,2,3,2,1,3,4,2,4,3,3,4,4,1,6,1,4,3,1,2,5,2,3,
%U 3,6,2,4,2,3,2,3,2,5,2,3,2,3,2,5,2,3,3,3,2,6
%N Number of r X s rectangles with squarefree side lengths such that r <= s, r + s = 2n and r | s.
%C a(n) is the number of divisors r of 2*n such that r <= n and r and n-r are squarefree. - _Robert Israel_, Apr 27 2020
%H Robert Israel, <a href="/A327567/b327567.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{i=1..n} mu(i)^2 * mu(2*n-i)^2 * (1-ceiling((2*n-i)/i)+floor((2*n-i)/i)), where mu is the Möbius function (A008683).
%e a(6) = 3; 2*6 = 12 has three rectangles with squarefree side lengths, 1 X 11, 2 X 10 and 6 X 6, such that 1 | 11, 2 | 10 and 6 | 6.
%p f:= proc(n) uses numtheory;
%p nops(select(r -> r <= n and issqrfree(r) and issqrfree(2*n-r), divisors(2*n),n))
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Apr 27 2020
%t Table[Sum[MoebiusMu[i]^2 MoebiusMu[2 n - i]^2 (1 - Ceiling[(2 n - i)/i] + Floor[(2 n - i)/i]), {i, n}], {n, 100}]
%Y Cf. A008683, A334361.
%K nonn,easy
%O 1,2
%A _Wesley Ivan Hurt_, Apr 24 2020
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