A308314 Decimal expansion of Sum_{k>=1} (1/A055642(k)^A055642(k)) where A055642(k) is the number of digits of the integer k.

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

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: A021151 A200116 A360500 * A097909 A143819 A372605

Adjacent sequences: A308311 A308312 A308313 * A308315 A308316 A308317

Keyword

nonn,base,cons

Author

Bernard Schott, May 19 2019

Status

approved