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

Author

Bernard Schott, May 19 2019

Keywords

• nonn
• base
• cons

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.

Examples

			168.05245375262168949085673320556724...
		

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.

Crossrefs

Cf. A055642, A138908.

Programs

Maple

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

PARI

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

Formula

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

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