Carefree Couple

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

Let  be the number of squarefree pairs,  the number of carefree couples,  the number of strongly carefree couples, and  the number of weakly squarefree couples with , illustrated above.

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

			(1)
			(2)

where  is the totient summatory function,  is the floor function, and  is the Möbius function.

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

Then

			(3)
			(4)
			(5)
			(6)

where the carefree and strongly carefree constants are given by

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

(OEIS A065464, A065473, and A118261; Moree 2005), where  is the Riemann zeta function.

EFERENCES:

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. https://arxiv.org/abs/math.NT/0510003.

Niklasch, G. "Some Number-Theoretical Constants." https://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."