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Khinchin's means $K_p$
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Numbers
mean 
value
$K_0$:
2.685452001065306445309714835481795693820382293994462953051152345557218859537152002801141174931847698
comment: Khinchin's constant, the geometric mean of the partial quotients [7].
$K_{-1}$:
1.745405662407346863494596309683661067294936618777984256595013773516078575220873425652057886456783242
comment: The Khinchin harmonic mean [8] [6].
$K_{-2}$:
1.450340328495630406052983076680697881408299979605904182171749071751762813556868765984131420829341887
comment: The Khinchin mean of order $-2$ [9].
$K_{-3}$:
1.313507078687985766717339447072786828158129861484792058809849805423881360338815925052429154118220861
comment: The Khinchin mean of order $-3$ [10].
$K_{-4}$:
1.236961809423730052626227244453422567420241131548937130091959279944265904948910655077042708923641281
comment: The Khinchin mean of order $-4$ [11].
$K_{-5}$:
1.189003926465513154062363732771403397386092512639671626409843693564665007968423061774246893333976312
comment: The Khinchin mean of order $-5$ [12].
$K_{-6}$:
1.156552374421514423152605998743410046840213070718761292231067077382867227150357664392651797183929343
comment: The Khinchin mean of order $-6$ [13].
$K_{-7}$:
1.133323363950865794910289694908868363599098282411797752596130817944257419876267445010354458049002027
comment: The Khinchin mean of order $-7$ [14].
$K_{-8}$:
1.115964408978716690619156419345349695769491182230400937063533110033709575050203118915011396467151700
comment: The Khinchin mean of order $-8$ [15].
$K_{-9}$:
1.102543136670728013836093402522568351022221284149318402331319847686325626947446033364139207872371785
comment: The Khinchin mean of order $-9$ [16].
$K_{-10}$:
1.091877041209612678276110979477638256493272651429656272710754913569735139707298906336665306654494092
comment: The Khinchin mean of order $-10$ [17].
$K_{1/2}$:
4.533095114977819974426915402241839871037325534446443652186233205238910862107216123585617209862597257
comment: The Khinchin mean of order $1/2$.
$K_{-1/2}$:
2.054840868357144293219335643846425321544225521250575579146975981797084500557366774744332307506997802
comment: The Khinchin mean of order $-1/2$.
Definition
Let $x=[a_0;a_1,a_2,\ldots]$ be the regular continued fraction expansion of a real number [4]. For $p<1$ the power mean of order $p$ of the partial quotients has, for almost every $x$, a limit that does not depend on $x$: $K_p=\lim_{n\to\infty}M_p(a_1,\ldots,a_n)$, where $M_p$ is the power mean and $M_0$ the geometric mean [2]. This table gives $K_p$ for the orders $p$ listed. The order $p=0$ gives Khinchin's constant.
Parameters
mean
—   the mean
Formulas
(1)
$\Pr(a_i=k)=\log_2\left(1+\frac{1}{k(k+2)}\right)$ for $k\geq1$.
(2)
$K_p=\left(\sum_{k=1}^{\infty} k^p\log_2\left(1+\frac{1}{k(k+2)}\right)\right)^{1/p}$ for $p<1$, $p\ne0$, and $K_0=\exp\left(\sum_{k=1}^{\infty} \log k\,\log_2\left(1+\frac{1}{k(k+2)}\right)\right)$.
(3)
$\log K_0=\frac{1}{\log 2}\sum_{n=1}^{\infty} \frac{\zeta(2n)-1}{n}\sum_{j=1}^{2n-1}\frac{(-1)^{j+1}}{j}$ [1].
(4)
For $p<1$, $p\ne0$, $K_p^p=\frac{1}{\log 2}\sum_{n=2}^{\infty} c_n(p)\left(\zeta(n-p)-1\right)$, where $c_n(p)=\sum_{j=1}^{\lfloor n/2\rfloor} \frac{(-1)^{n-2j}}{j}{p\choose n-2j}$.
Comments
(5)
Under the Gauss measure, whose density is $\frac{1}{(1+x)\log 2}$ and which is invariant under the Gauss map $x\mapsto 1/x-\lfloor 1/x\rfloor$ [3], the probability that a partial quotient is $k$ is (1). The constants $K_p$ are the power means of this distribution.
(6)
The row $K_0$ is Khinchin's constant [2] [5], and $K_{-1}$ is the Khinchin harmonic mean [6]. The arithmetic mean $K_1$ is infinite, and $K_p$ is finite exactly for $p<1$, so the table stops there rather than by choice.
(8)
The power mean increases with its order, so $K_{-10}<K_{-9}<\cdots<K_{-1}<K_{-1/2}<K_0<K_{1/2}$.
(9)
(3) and (4) write the means in values of the Riemann zeta function $\zeta$, which is what makes them quick to compute: $\zeta(n)-1$ falls off like $2^{-n}$, so the series needs a few hundred terms for a hundred digits where the product over $k$ in (2) would need more than any machine will do.
Programs
(P1)
Python
from mpmath import mp
mp.dps = 80
prob = lambda k: mp.log(1 + 1/(k*(k + 2))) / mp.log(2)
mp.exp(mp.nsum(lambda n: (mp.zeta(2*n) - 1)/n
               * mp.fsum((-1)**(j+1)/mp.mpf(j)
                         for j in range(1, 2*int(n))),
               [1, mp.inf]) / mp.log(2))                     # K_0
mp.nsum(lambda k: prob(k)/k, [1, mp.inf]) ** -1              # K_{-1}
References
[1]
D. H. Bailey, J. M. Borwein and R. E. Crandall, On the Khintchine constant, Mathematics of Computation 66 (1997), 417-431. (doi)
Links
Similar tables
Lévy's constant —   the other almost-sure limit of the regular continued fraction, taken over the denominators of the convergents rather than over the partial quotients
Lochs's constant —   how many partial quotients a decimal digit determines, for almost every real number
Values of the Riemann zeta function at rational numbers —   the zeta values $\zeta(2n)$ and $\zeta(n-p)$ in (3) and (4)
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds $K_p$ for $p=0,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10,\frac12,-\frac12$, and $K_p$ is finite for every $p<1$)
How they were obtained:

Each row was computed from (3) for $p=0$ and from (4) for $p\ne0$, twice: once at $150$ working decimal digits with $680$ grouped zeta terms, and once at $210$ working decimal digits with $920$ grouped zeta terms.

more

The two values agreed in their first $105$ significant digits, and what is stored is the union of the two decimal values. Agreement between two truncations is evidence and not a proof, which is what the rigour says. The rows $K_0,K_{-1},\ldots,K_{-10}$ were compared with the decimal expansions in the OEIS entries named in their comments and with the table on MathWorld's Khinchin harmonic mean page [6]. The two half-integer rows have no OEIS entry to compare with and were checked against direct partial sums of (2), with the positive tail bounded by $\sum_{k>N}k^{p-2}/\log 2$; at $N=200000$ the resulting intervals contain the stored values. The monotonicity in (8) was checked on the stored values.



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