Carefree Couple

Define a carefree couple as a pair of positive integers (a,b) such that a and b are relatively prime (i.e., GCD(a,b)=1) and a is squarefree. Similarly, define a strongly carefree couple as a pair (a,b) such that GCD(a,b)=1 and both a and b are squarefree, and a weakly carefree couple as a pair (a,b) such that GCD(a,b)=1 and at least of one a and b is squarefree.


Let C_0(x) be the number of squarefree pairs, C_1(x) the number of carefree couples, C_2(x) the number of strongly carefree couples, and C_3(x) the number of weakly squarefree couples with a,b<=x, illustrated above.

The numbers of squarefree pairs C_0(n) for n=1, 2, ... are 1, 3, 7, 11, 19, 23, 35, 43, 55, ... (OEIS A018805), which has closed forms


where Phi(n) is the totient summatory function, |_x_| is the floor function, and mu(n) is the Möbius function.

The numbers of carefree couples C_1(n) for n=1, 2, ... are 1, 3, 7, 9, 16, 20, 31, 35, 39, ... (OEIS A118258); the numbers of strongly carefree couples C_2(n) are 1, 3, 7, 7, 13, 17, 27, 27, ... (OEIS A118259); and the numbers of weakly carefree couples C_3(n) are 1, 3, 7, 11, 19, 23, 35, 43, 51, ... (OEIS A118260).



where the carefree and strongly carefree constants are given by


(OEIS A065464, A065473, and A118261; Moree 2005), where zeta(2)=pi^2/6 is the Riemann zeta function.

See also

Prime Products, Squarefree

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Finch, S. R. "Carefree Couples." §2.5.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 110-112, 2003.Moree, P. "Counting Carefree Couples." 30 Sep 2005., G. "Some Number-Theoretical Constants.", M. R. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. New York: Springer-Verlag, p. 54, 1997.Sloane, N. J. A. Sequences A015614, A018805, A065464, A065473, A118258, A118259, A118260, and A118261 in "The On-Line Encyclopedia of Integer Sequences."

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Carefree Couple

Cite this as:

Weisstein, Eric W. "Carefree Couple." From MathWorld--A Wolfram Web Resource.

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