A118259 Numbers of strongly carefree couples (a,b) with a,b <= n.

1, 3, 7, 7, 13, 17, 27, 27, 27, 33, 47, 47, 63, 73, 85, 85, 107, 107, 131, 131, 147, 165, 195, 195, 195, 215, 215, 215, 249, 265, 303, 303, 329, 355, 385, 385, 431, 461, 495, 495, 547, 569, 625, 625, 625, 661, 721, 721, 721, 721, 763, 763, 827, 827, 877, 877

Offset: 1

Author

Eric W. Weisstein, Apr 20 2006

Keywords

• nonn

Comments

(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

References

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

Links

• Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
• Eric Weisstein's World of Mathematics, Carefree Couple.

Crossrefs

Cf. A005117, A065473, A118258, A118260.

Programs

Mathematica

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 *)

PARI

a(n)=sum(i=1,n,sum(j=1,n,moebius(i*j)^2)) \\ Benoit Cloitre, Oct 10 2009

Formula

a(n) = Sum_{i,j=1...n} mu(i*j)^2. - Benoit Cloitre, Oct 10 2009

From Amiram Eldar, Mar 03 2021: (Start)

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

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