OFFSET
1,2
COMMENTS
Represents the value of the Dirichlet series for A011655 (principal Dirichlet character mod 3) at s=2.
Equals the asymptotic mean of the abundancy index of the numbers that are not divisible by 3 (A001651). - Amiram Eldar, May 12 2023
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
R. J. Mathar, Table of Dirichlet L-Series, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
FORMULA
Equals (4/3)*A100044.
Equals Sum_{n>=0} (1/(3*n+1)^2 + 1/(3*n+2)^2).
From Peter Luschny, May 13 2020: (Start)
Equals (8/9) * Sum_(k>=1) 1/k^2 =8/9 *A013661.
Equals -(16/9) * Sum_(k>=1) (-1)^k/k^2 = -16/9 * A072691.
Equals (64/27) * ( Integral_{x=0..1} sqrt(1 - x^2) )^2 = 64/27 * A091476. (End)
Equals Integral_{x=0..oo} log(x)/(x^3 - 1) dx. - Amiram Eldar, Aug 12 2020
EXAMPLE
1.4621636149762012768643690370186...
MAPLE
evalf(4*Pi^2/27) ;
MATHEMATICA
RealDigits[(4Pi^2)/27, 10, 120][[1]] (* Harvey P. Dale, Dec 20 2012 *)
PROG
(PARI) 4*Pi^2/27 \\ G. C. Greubel, Mar 08 2018
(Magma) R:= RealField(); 4*Pi(R)^2/27; // G. C. Greubel, Mar 08 2018
(Magma) R:=RealField(106); SetDefaultRealField(R); n:=4*Pi(R)^2/27; Reverse(Intseq(Floor(10^105*n))); // Bruno Berselli, Mar 13 2018
(Julia)
using Nemo
R = RealField(310)
t = const_pi(RR) + const_pi(RR); s = t * t
s / RR(27) |> println # Peter Luschny, Mar 13 2018
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Jul 20 2012
STATUS
approved