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A073668 Decimal expansion of Sum_{k=1..inf} 1/(10^k-1). 14
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Parallels A000005 up to a(46).

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

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..104.

FORMULA

From Eric Desbiaux, Mar 11 2009: (Start)

Equals Sum_{k >= 1}, 1/((2^k*5^k)-1).

Equals Sum_{k >= 1}, (1/2^k)*(1/5^k)/(1-((1/2^k)*(1/5^k))).

Sum_{k >= 1},1/(5^k) = 1/4.

Sum_{k >= 1},1/(2^k) = 1.

Sum_{k >= 1},(1/5^k)/(1-((1/2^k)*(1/5^k))) = 0.2726344339156...

Sum_{k >= 1},(1/2^k)/(1-((1/2^k)*(1/5^k))) = 1.0582125127815...

Sum_{k >= 1}, 1/(1-((1/2^k)*(1/5^k))) - 1 = A073668.

(End)

EXAMPLE

0.122324243426244526264428344628264449244... = A065444/9.

MATHEMATICA

RealDigits[ N[ Sum[1/(10^k - 1), {k, 1, Infinity}], 120]] [[1]]

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

PROG

(PARI) suminf(k=1, 1/(10^k-1)) \\ Charles R Greathouse IV, Oct 05 2014

CROSSREFS

Sequence in context: A000005 A122667 A122668 * A302051 A066800 A218705

Adjacent sequences:  A073665 A073666 A073667 * A073669 A073670 A073671

KEYWORD

cons,nonn

AUTHOR

Robert G. Wilson v, Aug 29 2002

STATUS

approved

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Last modified September 25 20:53 EDT 2018. Contains 315425 sequences. (Running on oeis4.)