A334362 Number of r X s rectangles with integer sides such that r <= s, r + s = 2n and s/r is squarefree.

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

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

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

Adjacent sequences: A334359 A334360 A334361 * A334363 A334364 A334365

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 24 2020

Status

approved