A118258 Numbers of carefree couples (a,b) with a,b<=n.

1, 3, 7, 9, 16, 20, 31, 35, 39, 46, 63, 67, 87, 98, 112, 119, 146, 152, 182, 189, 209, 228, 265, 273, 286, 308, 321, 330, 375, 391, 440, 453, 486, 515, 554, 565, 624, 657, 698, 712, 778, 801, 871, 888, 906, 946, 1022, 1037, 1063, 1080, 1133, 1152, 1236, 1252

Offset: 1

Author

Eric W. Weisstein, Apr 20 2006

Keywords

• nonn

Comments

(a, b) is a carefree couple if gcd(a, b) = 1 and a is squarefree (A005117). - Amiram Eldar, Mar 03 2021

Examples

			a(4) = 9 because there are 9 ordered pairs (i,j) of positive integers such that 1<=i,j<=4, gcd(i,j)=1 and i is squarefree: (1,1), (2,1), (3,1), (1,2), (3,2), (1,3), (2,3), (1,4), (3,4). - _Geoffrey Critzer_, Jan 12 2015
		

References

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..100000
• Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.
• Eric Weisstein's World of Mathematics, Carefree Couple.

Crossrefs

Cf. A005117, A065464, A118259, A118260.

Programs

Maple

F:= proc(n) local A,a;
  A:= select(numtheory:-issqrfree, [$1..n]);
  add(nops(select(y->igcd(a,y)=1, [$1..n])),a=A);
end proc:
seq(F(n),n=1..100); # Robert Israel, Jan 12 2015

Mathematica

Table[nn = n;Select[Level[Table[Table[{i, j}, {i, 1, nn}], {j, 1, nn}], {2}], Apply[GCD, #] == 1 && SquareFreeQ[#[[1]]] &] // Length, {n, 1, 54}] (* Geoffrey Critzer, Jan 12 2015 *)

PARI

a(n)=my(s); forsquarefree(m=1,n, s+=sumdiv(m, d, n\d*moebius(d))); s \\ Charles R Greathouse IV, Jan 25 2018

Formula

From Amiram Eldar, Mar 03 2021: (Start)

a(n) = (A118259(n) + A118260(n))/2.

a(n) ~ A065464 * n^2 + O(n*log(n)). (End)