A214369 -id:A214369 - cp's OEIS frontend

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

A.H.M. Smeets, Nov 20 2018

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).

			1.364307005210476133522526372453248019298380496653806838456515694...

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

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

Cf. A000005.

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

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

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

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

A329064 Decimal expansion of Sum_{n>=1} 1/(3^n-2^n).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 03 2019

			1.2749815155811420993...

Cf. A065442 (Sum_{n>=1} 1/(2^n-1)), A214369 (Sum_{n>=1} 1/(3^n-1)).

Cf. A001047 (3^n-2^n).

RealDigits[Total[Table[1/(3^n-2^n),{n,200}]],10,100][[1]] (* Harvey P. Dale, Nov 27 2024 *)

default(realprecision, 100); suminf(n=1, 1/(3^n-2^n))

cp's OEIS Frontend

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

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