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A308314 Decimal expansion of Sum_{k>=1} (1/A055642(k)^A055642(k)) where A055642(k) is the number of digits of the integer k. 2
1, 6, 8, 0, 5, 2, 4, 5, 3, 7, 5, 2, 6, 2, 1, 6, 8, 9, 4, 9, 0, 8, 5, 6, 7, 3, 3, 2, 0, 5, 5, 6, 7, 2, 4, 5, 2, 1, 9, 6, 5, 2, 6, 7, 9, 9, 7, 1, 9, 8, 4, 9, 5, 0, 4, 9, 1, 5, 5, 7, 0, 3, 5, 9, 8, 1, 4, 3, 7, 9, 8, 3, 4, 8, 1, 7, 5, 7, 0, 8, 8, 9, 4, 8, 3, 4, 6, 1, 6, 4, 4, 4, 5, 0, 7, 8, 4, 8, 6, 4 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

With summation by parts to obtain 1st formula:

Sum_{k>=1} (1/length(k)^length(k)) =

Sum_{m=1..9} (1/1^1) + Sum_{m=10..99} (1/2^2) + Sum_{m=100...999} (1/3^3) + Sum_{m=1000...9999} (1/4^4) + ... =

9*(1/1^1) + 90*(1/2^2) + 900*(1/3^3) + 9000*(1/4^4) + 90000*(1/5^5) + ... =

9 ( 1/1^1 + 10^1/2^2 + 10^2/3^3 + 10^3/4^4 + 10^4/5^5 + ... =

(9/10) * (10^1/1^1 + 10^2/2^2 + 10^3/3^3 + 10^4/4^4 + 10^5/5^5 + ... =

(9/10) * ( (10/1)^1 + (10/2)^2 + (10/3)^3 + (10/4)^4 + (10/5)^5 + ... =

(9/10) * Sum_{m>=1} (10/m)^m.

REFERENCES

Xavier Merlin, Methodix Analyse, Ellipses, 1997, Exercice 22 p. 120.

J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.1.h" p. 248.

LINKS

Table of n, a(n) for n=3..102.

FORMULA

Equals (9/10) * Sum_{k>=1} (10/k)^k.

Equals Sum_{n>=1} (1/A138908(n)).

EXAMPLE

168.05245375262168949085673320556724...

MAPLE

evalf((9/10) * Sum((10/n)^n, n=1..infinity), 100);

PROG

(PARI) (9/10) * suminf(k=1, (10/k)^k) \\ Michel Marcus, Jun 08 2019

CROSSREFS

Cf. A055642, A138908.

Sequence in context: A268508 A021151 A200116 * A097909 A143819 A021599

Adjacent sequences:  A308311 A308312 A308313 * A308315 A308316 A308317

KEYWORD

nonn,base,cons

AUTHOR

Bernard Schott, May 19 2019

STATUS

approved

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Last modified May 28 08:23 EDT 2022. Contains 354112 sequences. (Running on oeis4.)