The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A344428 Decimal expansion of exp(-2/5). 2
6, 7, 0, 3, 2, 0, 0, 4, 6, 0, 3, 5, 6, 3, 9, 3, 0, 0, 7, 4, 4, 4, 3, 2, 9, 2, 5, 1, 4, 7, 8, 2, 6, 0, 7, 1, 9, 3, 6, 9, 8, 0, 9, 2, 5, 2, 1, 0, 8, 1, 2, 1, 9, 9, 8, 8, 8, 9, 1, 0, 3, 3, 1, 6, 2, 5, 8, 9, 4, 1, 7, 5, 1, 2, 0, 3, 5, 3, 7, 4, 3, 8, 2, 6, 3, 3, 7, 5, 4, 3, 9 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Let f(s) = zeta(zeta(s+1)) - 1, where zeta(s) is the Riemann zeta function. Then f(s) is a strictly increasing function from (0, +oo) to (0, +oo), lim_{s->0+} f(s) = 0, lim_{s->+oo} f(s) = +oo.
Conjecture:
(i) f(s) has a unique fixed point s = A069995 - 1 in (0, +oo);
(ii) Lim_{s->+oo} f(s)/2^s = 1, lim_{s->0+} f(s)/2^(-1/s) = exp(-2/5) = A344428.
If these are true, let s_0 be any real number > alpha, s_n = zeta(s_{n-1}) for n >= 1, where alpha = A069995 is the fixed point of zeta(s) in (1, +oo), then {s_{2n}} diverges quickly to +oo, {s_{2n+1}} converges quickly to 1.
This is because the derivative of zeta(zeta(s)) - s at s = alpha is (zeta'(alpha))^2 - 1 = A344427^2 - 1 > 0, so (i) implies that zeta(zeta(s)) > s for s > alpha and zeta(zeta(s)) < s for 1 < s < alpha, hence ... > s_{2n} > s_{2n-2} > ... > s_2 > s_0 > alpha > s_1 > s_3 > ... > s_{2n+1} > ..., and it follows from (i) that lim_{n->+oo} s_{2n} = +oo, lim_{n->+oo} s_{2n+1} = 1. By definition s_n - 1 = f(s_{n-2} - 1), n >= 2. For large n, s_{2n} - 1 is approximately equal to 2^(s_{2(n-1)} - 1), and 1/(s_{2n+1} - 1) is approximately equal to exp(2/5) * 2^(1/(s_{2(n-1)+1} - 1)).
LINKS
EXAMPLE
exp(-2/5) = 0.67032004603563930074... In comparison, (zeta(zeta(0.001+1)) - 1) / 2^(-1/0.001) = 0.67022226725425164463...
MATHEMATICA
RealDigits[Exp[-2/5], 10, 100][[1]] (* Amiram Eldar, May 19 2021 *)
PROG
(PARI) default(realprecision, 100); exp(-2/5)
CROSSREFS
Sequence in context: A330157 A225962 A011098 * A224238 A227718 A198932
KEYWORD
nonn,cons
AUTHOR
Jianing Song, May 19 2021
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 17 19:53 EDT 2024. Contains 372607 sequences. (Running on oeis4.)