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

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, 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, 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

Offset

1,2

Comments

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).

Links

Table of n, a(n) for n=1..128.

Nobushige Kurokawa and Yuichiro Taguchi, A p-analogue of Euler’s constant and congruence zeta functions, Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 2 (2018), 13-16.

Eric Weisstein's World of Mathematics, Erdős-Borwein Constant

Eric Weisstein's World of Mathematics, Lambert Series

Formula

Equals 2*L(1/3) = 2 * A214369, where L is the Lambert series.

Equals 2 * Sum_{k>=1} x^(k^2)*(1+x^n)/(1-x^n) where x = 1/3.

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

Example

1.364307005210476133522526372453248019298380496653806838456515694...

Maple

evalf[130](sum(2/(3^k-1), k=1..infinity)); # Muniru A Asiru, Dec 20 2018

Mathematica

RealDigits[Sum[2/(3^k-1), {k, 1, Infinity}], 10, 120][[1]] (* Amiram Eldar, Nov 21 2018 *)

Prog

(PARI) suminf(k=1, 2/(3^k-1)) \\ Michel Marcus, Nov 20 2018

Crossrefs

Cf. A065442 (base 2), A321873 (base 4).

Cf. A000005.

Sequence in context: A359572 A011307 A243625 * A221363 A245943 A245557

Adjacent sequences: A321869 A321870 A321871 * A321873 A321874 A321875

Keyword

nonn,cons

Author

A.H.M. Smeets, Nov 20 2018

Status

approved