This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A070072 #36 May 12 2025 05:27:56
%S A070072 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,
%T A070072 108,108,108,125,133,152,152,165,178,193,193,216,231,248,248,274,285,
%U A070072 313,313,313,331,361,361,361,361,382,382,414
%N A070072 Number of distinct rectangles with integer sides <= n and squarefree area.
%H A070072 Reinhard Zumkeller, <a href="/A070072/b070072.txt">Table of n, a(n) for n = 1..500</a>
%H A070072 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>.
%F A070072 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
%F A070072 a(n) = (A118259(n) + 1)/2. - _Ridouane Oudra_, May 06 2025
%F A070072 a(n) = c * n^2 / 2 + O(n*log(n)), where c = Product_{p prime} (1 - (3*p-2)/(p^3)) (A065473). - _Amiram Eldar_, May 12 2025
%e A070072 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.
%o A070072 (Haskell)
%o A070072 a070072 n = length [() | x <- [1..n], y <- [1..x], a008966 (x*y) == 1]
%o A070072 -- _Reinhard Zumkeller_, May 26 2012
%o A070072 (Magma) [&+[&+[MoebiusMu(i*j)^2:j in [1..i]]:i in [1..n]]:n in [1..53]]; // _Marius A. Burtea_, Oct 17 2019
%Y A070072 Cf. A005117, A065473, A070073.
%Y A070072 Cf. A008966, A008683, A118259.
%K A070072 nonn
%O A070072 1,2
%A A070072 _Reinhard Zumkeller_, Apr 21 2002