Carefree Couple
المؤلف:
Finch, S. R.
المصدر:
"Carefree Couples." §2.5.1 in Mathematical Constants. Cambridge, England: Cambridge University Pres
الجزء والصفحة:
...
13-6-2020
1045
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
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
where the carefree and strongly carefree constants are given by
(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."
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