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

Author

A.H.M. Smeets, Nov 20 2018

Keywords

• nonn
• cons

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).

Examples

			1.364307005210476133522526372453248019298380496653806838456515694...
		

Links

• 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

Crossrefs

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

Cf. A000005.

Programs

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 *)

PARI

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

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