A214369 - OEIS

A214369

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

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

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OFFSET

0,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

FORMULA

Equals Sum_{n>=1} 1/A024023(n).

Equals Sum_{k>=1} d(k)/3^k, where d(k) is the number of divisors of k (A000005). - Amiram Eldar, May 17 2020

EXAMPLE

Equals 0.6821535026052380667...

MAPLE

evalf(sum(1/(3^k-1), k=1..infinity), 120); # Vaclav Kotesovec, Oct 18 2014

# second program with faster converging series

evalf( add( (1/3)^(n^2)*(1 + 2/(3^n - 1)), n = 1..14 ), 105); # Peter Bala, Jan 30 2022

MATHEMATICA

RealDigits[ NSum[1/(3^n - 1), {n, 1, Infinity}, WorkingPrecision -> 110, NSumTerms -> 100], 10, 105] // First (* or *) 1 - (Log[2] + QPolyGamma[0, 1, 1/3])/Log[3] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jun 05 2013 *)

x = 1/3; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* Robert G. Wilson v, Oct 12 2014 after an observation and the formula of Amarnath Murthy, see A073668 *)

PROG

(PARI) suminf(n=1, 1/(3^n-1)) \\ Michel Marcus, Mar 11 2017

CROSSREFS

Cf. A024023, A065442, A073668, A248721, A248722, A248723, A248724, A248725, A248726.

Sequence in context: A246844 A246845 A374815 * A242769 A189090 A075549

Adjacent sequences: A214366 A214367 A214368 * A214370 A214371 A214372

KEYWORD

cons,nonn

AUTHOR

R. J. Mathar, Jul 14 2012

EXTENSIONS

More terms from Jean-François Alcover, Feb 12 2013

STATUS

approved