A104141 - OEIS

%I #49 Apr 23 2024 10:07:14

%S 3,0,3,9,6,3,5,5,0,9,2,7,0,1,3,3,1,4,3,3,1,6,3,8,3,8,9,6,2,9,1,8,2,9,

%T 1,6,7,1,3,0,7,6,3,2,4,0,1,6,7,3,9,6,4,6,5,3,6,8,2,7,0,9,5,6,8,2,5,1,

%U 9,3,6,2,8,8,6,7,0,6,3,2,3,5,7,3,6,2,7,8,2,1,7,7,6,8,6,5,5,1,2,8

%N Decimal expansion of 3/Pi^2.

%C 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.

%C 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

%C 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

%D J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1995, p. 156

%D L. E. Dickson, History of the Theory of Numbers, Vol. I pp. 126 Chelsea NY 1966.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FareySequence.html">Farey Sequence</a>

%H Joshua Zelinsky, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Zelinsky/zelinsky9.html">Tau Numbers: A Partial Proof of a Conjecture and Other Results</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Sum_{n>=1} 1/A039956(n)^2. - _Amiram Eldar_, May 22 2020

%F From _Terry D. Grant_, Oct 31 2020: (Start)

%F Equals (-1)*zeta(0)/zeta(2).

%F Equals 1/(zeta(2)/2).

%F Equals 1/A195055.

%F Equals (1/2)*Sum_{k>=1} mu(k)/k^2.

%F (End)

%F From _Hugo Pfoertner_, Apr 23 2024: (Start)

%F Equals A059956/2.

%F Equals A082020/5. (End)

%e 3/Pi^2 = 0.303963550927013314331638389629...

%t l = RealDigits[N[3/Pi^2, 100]]; Prepend[First[l], Last[l]] (* _Ryan Propper_, Aug 04 2005 *)

%o (PARI) 3/Pi^2 \\ _Charles R Greathouse IV_, Mar 08 2013

%Y Cf. A000010, A002088, A000290.

%Y Cf. A046642, A030229, A030059, A039956, A276378.

%Y Cf. A013661, A195055, A306633, A082020, A088246.

%K nonn,cons

%O 0,1

%A _Lekraj Beedassy_, Mar 07 2005

%E More terms from _Ryan Propper_, Aug 04 2005