Carefree Couple

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

Let be the number of ordered pairs of relatively prime positive integers, the number of carefree couples, the number of strongly carefree couples, and the number of weakly carefree couples with , illustrated above.

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

			(1)
			(2)

for positive integers , 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).

As ,

			(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 2014), where is the Riemann zeta function, , and each prime product runs over all primes .