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Decimal expansion of Sum_{k>=1} 1/(8^k - 1).
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%I #24 Jan 30 2022 15:57:58

%S 1,6,0,9,6,6,1,8,4,3,1,5,0,6,2,3,9,6,8,0,5,3,0,2,5,6,4,1,4,3,6,4,2,8,

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

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

%N Decimal expansion of Sum_{k>=1} 1/(8^k - 1).

%H G. C. Greubel, <a href="/A248725/b248725.txt">Table of n, a(n) for n = 0..10000</a>

%F Equals Sum_{k>=1} d(k)/8^k, where d(k) is the number of divisors of k (A000005). - _Amiram Eldar_, Jun 22 2020

%e 0.16096618431506239680530256414364288555074385602532834636083591864782394085800...

%p evalf(sum(1/(8^k-1), k=1..infinity),120) # _Vaclav Kotesovec_, Oct 18 2014

%p # second program with faster converging series

%p evalf( add( (1/8)^(n^2)*(1 + 2/(8^n - 1)), n = 1..10), 105); # _Peter Bala_, Jan 30 2022

%t x = 1/8; RealDigits[ Sum[ DivisorSigma[0, k] x^k, {k, 1000}], 10, 105][[1]] (* after an observation and the formula of _Amarnath Murthy_, see A073668 *)

%o (PARI) suminf(k=1, 1/(8^k-1)) \\ _Michel Marcus_, Oct 18 2014

%Y Cf. A000005, A065442, A073668, A214369, A248721, A248722, A248723, A248724, A248726.

%K nonn,cons

%O 0,2

%A _Robert G. Wilson v_, Oct 12 2014