cp's OEIS Frontend

A073668 Decimal expansion of Sum_{k>=1} 1/(10^k - 1).

%I A073668 #49 Oct 29 2023 01:51:14
%S A073668 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,
%T A073668 4,9,2,4,4,8,2,8,2,6,6,4,3,0,3,6,4,6,2,8,4,8,4,4,3,2,2,4,6,7,4,8,2,6,
%U A073668 4,8,3,2,2,4,6,6,4,8,3,0,5,4,3,2,4,4,4,8,3,2,4,6,4,4,5,2,2,6,6,9,2,8,2,8,8
%N A073668 Decimal expansion of Sum_{k>=1} 1/(10^k - 1).
%C A073668 Parallels A000005 up to a(46).
%C A073668 Sum_{k>=1} x^k/(1-x^k) = Sum_{k>=1} tau(k)*x^k. Choosing x = 1/10 gives the result. - _Amarnath Murthy_, Oct 21 2002
%D A073668 Amarnath Murthy, Some interesting results on d(N), the number of divisors of a natural number, page 463, Octogon Mathematical Magazine, Vol. 8 No. 2, October 2000.
%F A073668 From _Eric Desbiaux_, Mar 11 2009: (Start)
%F A073668 Equals Sum_{k >= 1} 1/((2^k*5^k)-1).
%F A073668 Equals Sum_{k >= 1} (1/2^k)*(1/5^k)/(1-((1/2^k)*(1/5^k))).
%F A073668 Sum_{k >= 1} 1/(5^k) = 1/4.
%F A073668 Sum_{k >= 1} 1/(2^k) = 1.
%F A073668 Sum_{k >= 1} (1/5^k)/(1-((1/2^k)*(1/5^k))) = 0.2726344339156...
%F A073668 Sum_{k >= 1} (1/2^k)/(1-((1/2^k)*(1/5^k))) = 1.0582125127815...
%F A073668 Sum_{k >= 1} 1/(1-((1/2^k)*(1/5^k))) - 1 = A073668.
%F A073668 (End)
%F A073668 Fast computation via Lambert series: 0.122324243426... = Sum_{n>=1} x^(n^2)*(1+x^n)/(1-x^n) where x=1/10. - _Joerg Arndt_, Oct 18 2020
%e A073668 0.122324243426244526264428344628264449244... = A065444/9.
%p A073668 evalf(Sum(1/(10^k - 1), k = 1..infinity), 200) # _Vaclav Kotesovec_, Jul 16 2019
%p A073668 # second program with faster converging series after _Joerg Arndt_
%p A073668 evalf( add( (1/10)^(n^2)*(1 + 2/(10^n - 1)), n = 1..8), 105); # _Peter Bala_, Jan 30 2022
%t A073668 RealDigits[ N[ Sum[1/(10^k - 1), {k, 1, Infinity}], 120]] [[1]]
%o A073668 (PARI) suminf(k=1,1/(10^k-1)) \\ _Charles R Greathouse IV_, Oct 05 2014
%Y A073668 Cf. A065442, A214369, A248721, A248722, A248723, A248724, A248725, A248726.
%K A073668 cons,nonn
%O A073668 0,2
%A A073668 _Robert G. Wilson v_, Aug 29 2002