A070072 Number of distinct rectangles with integer sides <= n and squarefree area.

1, 2, 4, 4, 7, 9, 14, 14, 14, 17, 24, 24, 32, 37, 43, 43, 54, 54, 66, 66, 74, 83, 98, 98, 98, 108, 108, 108, 125, 133, 152, 152, 165, 178, 193, 193, 216, 231, 248, 248, 274, 285, 313, 313, 313, 331, 361, 361, 361, 361, 382, 382, 414

Offset

1,2

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Eric Weisstein's World of Mathematics, Squarefree.

Formula

a(n) = Sum_{i=1..n} Sum_{j= 1..i} mu(i*j)^2, where mu is the Moebius function (A008683). - Ridouane Oudra, Oct 17 2019

Example

There are seven rectangles with sides <= 5 having a squarefree area: 1 X 1, 1 X 2, 1 X 3, 1 X 5, 2 X 3, 2 X 5 and 3 X 5, whereas 1 X 4, 2 X 2, 2 X 4, 3 X 3, 3 X 4, 4 X 4, 4 X 5 and 5 X 5 are not squarefree; therefore a(5)=7.

Prog

(Haskell)
a070072 n = length [() | x <- [1..n], y <- [1..x], a008966 (x*y) == 1]
-- Reinhard Zumkeller, May 26 2012
(Magma) [&+[&+[MoebiusMu(i*j)^2:j in [1..i]]:i in [1..n]]:n in [1..53]]; // Marius A. Burtea, Oct 17 2019

Crossrefs

Cf. A005117, A070073.

Cf. A008966, A008683.

Sequence in context: A340448 A241387 A284612 * A265259 A238493 A362937

Adjacent sequences: A070069 A070070 A070071 * A070073 A070074 A070075

Keyword

nonn

Author

Reinhard Zumkeller, Apr 21 2002

Status

approved