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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.

CarefreeCouples

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

C_0(x)=2Phi(n)-1
(1)
=sum_(k=1)^(n)|_n/k_|^2mu(k)
(2)

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).

Then

C_1(x)=K_1x^2+O(xlnx)
(3)
C_2(x)=K_2x^2+O(xln^3x)
(4)
C_3(x)=2C_1(x)-C_2(x)
(5)
=K_3x^2+...,
(6)

where the carefree and strongly carefree constants are given by

K_1=1/([zeta(2)]^2)[1+1/((p+1)(p^2-1))]
(7)
=1/(zeta(2))product_(p)[1-1/(p(p+1))]
(8)
=product_(p)(1-(2p-1)/(p^3))
(9)
=0.4282495056770944...
(10)
K_2=1/([zeta(2)]^3)product_(p)[1+(2p+1)/((p+1)^2(p^2-1))]
(11)
=1/([zeta(2)]^2)product_(p)[1-1/((p+1)^2)]
(12)
=1/(zeta(2))product_(p)[1-2/(p(p+1))]
(13)
=product_(p)(1-1/p)^2(1+2/p)
(14)
=product_(p)(1-(3p-2)/(p^3))
(15)
=0.2867474284344787...
(16)
K_3=2K_1-K_2
(17)
=0.5697515...
(18)

(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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References

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. http://arxiv.org/abs/math.NT/0510003.Niklasch, G. "Some Number-Theoretical Constants." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml.Schroeder, 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."

Referenced on Wolfram|Alpha

Carefree Couple

Cite this as:

Weisstein, Eric W. "Carefree Couple." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CarefreeCouple.html

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