OFFSET
1,2
COMMENTS
(a, b) is a carefree couple if gcd(a, b) = 1 and a is 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
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.
FORMULA
From Amiram Eldar, Mar 03 2021: (Start)
a(n) ~ A065464 * n^2 + O(n*log(n)). (End)
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Apr 20 2006
STATUS
approved