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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 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.)