%I #16 Mar 03 2021 04:20:45

%S 1,3,7,7,13,17,27,27,27,33,47,47,63,73,85,85,107,107,131,131,147,165,

%T 195,195,195,215,215,215,249,265,303,303,329,355,385,385,431,461,495,

%U 495,547,569,625,625,625,661,721,721,721,721,763,763,827,827,877,877

%N Numbers of strongly carefree couples (a,b) with a,b <= n.

%C (a, b) is a strongly carefree couple if gcd(a, b) = 1 and both a and b are squarefree (A005117). - _Amiram Eldar_, Mar 03 2021

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.5.1 Carefree Couples, p. 110.

%H Pieter Moree, <a href="https://arxiv.org/abs/math/0510003">Counting carefree couples</a>, arXiv:math/0510003 [math.NT], 2005-2014.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarefreeCouple.html">Carefree Couple</a>.

%F a(n) = Sum_{i,j=1...n} mu(i*j)^2. - _Benoit Cloitre_, Oct 10 2009

%F From _Amiram Eldar_, Mar 03 2021: (Start)

%F a(n) = 2*A118258(n) - A118260(n).

%F a(n) ~ A065473 * n^2 + O(n*log(n)). (End)

%t Table[nn = n;Length[Select[Level[Table[Table[{i, j}, {i, 1, nn}], {j, 1, nn}], {2}],Apply[GCD, #] == 1 && SquareFreeQ[#[[1]]] &&SquareFreeQ[#[[2]]] &]], {n, 1, 56}] (* _Geoffrey Critzer_, Jan 13 2015 *)

%o (PARI) a(n)=sum(i=1,n,sum(j=1,n,moebius(i*j)^2)) \\ _Benoit Cloitre_, Oct 10 2009

%Y Cf. A005117, A065473, A118258, A118260.

%K nonn

%O 1,2

%A _Eric W. Weisstein_, Apr 20 2006