A327567 Number of r X s rectangles with squarefree side lengths such that r <= s, r + s = 2n and r | 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, 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, 3, 6, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 3, 3, 2, 6

Offset

1,2

Comments

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

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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).

Example

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.

Maple

f:= proc(n) uses numtheory;
  nops(select(r -> r <= n and issqrfree(r) and issqrfree(2*n-r), divisors(2*n), n))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 27 2020

Mathematica

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}]

Crossrefs

Cf. A008683, A334361.

Sequence in context: A194290 A329028 A257079 * A260372 A037180 A241675

Adjacent sequences: A327564 A327565 A327566 * A327568 A327569 A327570

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 24 2020

Status

approved