OFFSET
0,1
COMMENTS
3/Pi^2 is the limit of (Sum_{k=1..n} phi(k))/n^2, where phi(k) is Euler's totient A000010(k), i.e., of A002088(n)/A000290(n) as n tends to infinity.
The previous comment in the context of Farey series means that the length of the n-th Farey series can be approximated by multiplying this constant by n^2, "and that the approximation gets proportionally better as n gets larger", according to Conway and Guy. - Alonso del Arte, May 28 2011
The asymptotic density of the sequences of squarefree numbers with even number of prime factors (A030229), odd number of prime factors (A030059), and coprime to 6 (A276378). - Amiram Eldar, May 22 2020
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 156
L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.
LINKS
Eric Weisstein's World of Mathematics, Farey Sequence
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.
FORMULA
Equals Sum_{n>=1} 1/A039956(n)^2. - Amiram Eldar, May 22 2020
From Terry D. Grant, Oct 31 2020: (Start)
Equals (-1)*zeta(0)/zeta(2).
Equals 1/(zeta(2)/2).
Equals 1/A195055.
Equals (1/2)*Sum_{k>=1} mu(k)/k^2.
(End)
From Hugo Pfoertner, Apr 23 2024: (Start)
Equals A059956/2.
Equals A082020/5. (End)
EXAMPLE
3/Pi^2 = 0.303963550927013314331638389629...
MATHEMATICA
l = RealDigits[N[3/Pi^2, 100]]; Prepend[First[l], Last[l]] (* Ryan Propper, Aug 04 2005 *)
PROG
(PARI) 3/Pi^2 \\ Charles R Greathouse IV, Mar 08 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Lekraj Beedassy, Mar 07 2005
EXTENSIONS
More terms from Ryan Propper, Aug 04 2005
STATUS
approved