A334361 Number of r X s rectangles with squarefree side lengths such that r < s, r + s = 2n and r | s.

0, 1, 1, 2, 0, 2, 1, 2, 2, 2, 1, 3, 0, 2, 1, 2, 1, 4, 1, 4, 3, 3, 0, 3, 0, 2, 2, 2, 1, 4, 1, 1, 2, 3, 2, 4, 1, 2, 2, 3, 0, 5, 1, 4, 3, 2, 1, 3, 2, 1, 2, 4, 1, 4, 2, 3, 3, 3, 0, 6, 0, 3, 3, 1, 1, 4, 1, 3, 2, 5, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 6

Offset

1,4

Links

Table of n, a(n) for n=1..90.

Formula

a(n) = Sum_{i=1..n-1} (1 - ceiling((2*n-i)/i) + floor((2*n-i)/i)) * mu(i)^2 * mu(2*n-i)^2, where mu is the Möbius function (A008683).

Example

a(12) = 3; 2*12 = 24 has 3 rectangles with squarefree side lengths, 1 X 23, 2 X 22 and 3 X 21 such that 1 | 23, 2 | 22 and 3 | 21.

Mathematica

Table[Sum[MoebiusMu[i]^2 MoebiusMu[2 n - i]^2 (1 - Ceiling[(2 n - i)/i] + Floor[(2 n - i)/i]), {i, n - 1}], {n, 100}]

Crossrefs

Cf. A008683, A327567.

Sequence in context: A319514 A141454 A301564 * A360568 A219791 A029345

Adjacent sequences: A334358 A334359 A334360 * A334362 A334363 A334364

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 24 2020

Status

approved