cp's OEIS Frontend

A321872 Decimal expansion of the sum of reciprocals of repunit numbers base 3, Sum_{k>=1} 2/(3^k - 1).

%I A321872 #34 Feb 16 2025 08:33:57
%S A321872 1,3,6,4,3,0,7,0,0,5,2,1,0,4,7,6,1,3,3,5,2,2,5,2,6,3,7,2,4,5,3,2,4,8,
%T A321872 0,1,9,2,9,8,3,8,0,4,9,6,6,5,3,8,0,6,8,3,8,4,5,6,5,1,5,6,9,4,2,7,3,5,
%U A321872 4,3,6,6,9,5,4,8,3,5,7,4,6,5,8,0,1,9,2,4,2,5,3,8,0,6,0,9,0,6,6,2,7,5,0,0,6,4,9,9,6,1,4,3,9,7,3,4,5,1,7,8,8,1,5,5,0,8,3,2
%N A321872 Decimal expansion of the sum of reciprocals of repunit numbers base 3, Sum_{k>=1} 2/(3^k - 1).
%C A321872 The sums of reciprocal repunit numbers are related to the Lambert series. A special case is the sum of repunit numbers in base 2, which is known as the Erdős-Borwein constant (A065442).
%H A321872 Nobushige Kurokawa and Yuichiro Taguchi, <a href="http://dx.doi.org/10.3792/pjaa.94.13">A p-analogue of Euler’s constant and congruence zeta functions</a>, Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 2 (2018), 13-16.
%H A321872 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Erdos-BorweinConstant.html">Erdős-Borwein Constant</a>
%H A321872 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertSeries.html">Lambert Series</a>
%F A321872 Equals 2*L(1/3) = 2 * A214369, where L is the Lambert series.
%F A321872 Equals 2 * Sum_{k>=1} x^(k^2)*(1+x^n)/(1-x^n) where x = 1/3.
%F A321872 Equals 2*Sum_{m>=1} tau(m)/3^m where tau(m) is A000005(m), the number of divisors of m. - _Michel Marcus_, Mar 18 2019
%e A321872 1.364307005210476133522526372453248019298380496653806838456515694...
%p A321872 evalf[130](sum(2/(3^k-1),k=1..infinity)); # _Muniru A Asiru_, Dec 20 2018
%t A321872 RealDigits[Sum[2/(3^k-1), {k, 1, Infinity}], 10, 120][[1]] (* _Amiram Eldar_, Nov 21 2018 *)
%o A321872 (PARI) suminf(k=1, 2/(3^k-1)) \\ _Michel Marcus_, Nov 20 2018
%Y A321872 Cf. A065442 (base 2), A321873 (base 4).
%Y A321872 Cf. A000005.
%K A321872 nonn,cons
%O A321872 1,2
%A A321872 _A.H.M. Smeets_, Nov 20 2018