en.wikipedia.org/wiki/MRB_constant

Here are some record computations. If you know of any others let
me know.

On or about Dec 31, 1998 I computed 1 digit of the (additive
inverse of the) MRB constant with my TI-92's, by adding 1-sqrt(2)
+3^(1/3)-4^(1/4) as far as I could and then by using the sum
feature to compute ∑1000n=1(−1)n(n1/n). That first digit, by the
way, is just 0.

On Jan 11, 1999 I computed 3 digits of the MRB constant with the
Inverse Symbolic Calculator.

In Jan of 1999 I computed 4 correct digits of the MRB constant
using Mathcad 3.1 on a 50 MHz 80486 IBM 486 personal computer
operating on Windows 95.

Shortly afterwards I computed 9 correct digits of the MRB constant
using Mathcad 7 professional on the Pentium II mentioned below.

On Jan 23, 1999 I computed 500 digits of the MRB constant with
the online tool called Sigma.

In September of 1999, I computed the first 5,000 digits of the MRB
Constant on a 350 MHz Pentium II with 64 Mb of ram using the
simple PARI commands \p 5000;sumalt(n=1,((-1)^n*(n^(1/n)-1))),
after allocating enough memory.

On June 10-11, 2003 over a period, of 10 hours, on a 450mh P3
with an available 512mb RAM, I computed 6,995 accurate digits of
the MRB constant.

Using a Sony Vaio P4 2.66 GHz laptop computer with 960 MB of
available RAM, on 2:04 PM 3/25/2004, I finished computing 8000
digits of the MRB constant.

On March 01, 2006 with a 3GH PD with 2GBRAM available, I
computed the first 11,000 digits of the MRB Constant.

On Nov 24, 2006 I computed 40, 000 digits of the MRB Constant in
33hours and 26min via my own program in written in Mathematica
5.2. The computation was run on a 32-bit Windows 3GH PD
desktop computer using 3.25 GB of Ram.

Finishing on July 29, 2007 at 11:57 PM EST, I computed 60,000
digits of MRB Constant. Computed in 50.51 hours on a 2.6 GH
AMD Athlon with 64 bit Windows XP. Max memory used was 4.0
GB of RAM.

Finishing on Aug 3 , 2007 at 12:40 AM EST, I computed 65,000
digits of MRB Constant. Computed in only 50.50 hours on a 2.66
GH Core2Duo using 64 bit Windows XP. Max memory used was
5.0 GB of RAM.

Finishing on Aug 12, 2007 at 8:00 PM EST, I computed 100,000
digits of MRB Constant. They were computed in 170 hours on a
2.66GH Core2Duo using 64 bit Windows XP. Max memory used
was 11.3 GB of RAM. Median (typical) daily record of memory used
was 8.5 GB of RAM.

Finishing on Sep 23, 2007 at 11:00 AM EST, I computed 150,000
digits of MRB Constant. They were computed in 330 hours on a
2.66GH Core2Duo using 64 bit Windows XP. Max memory used
was 22 GB of RAM. Median (typical) daily record of memory used
was 17 GB of RAM.

Finishing on March 16, 2008 at 3:00 PM EST, I computed 200,000
digits of MRB Constant using Mathematica 5.2. They were
computed in 845 hours on a 2.66GH Core2Duo using 64 bit
Windows XP. Max memory used was 47 GB of RAM. Median
(typical) daily record of memory used was 28 GB of RAM.

Washed away by Hurricane Ike -- on September 13, 2008
sometime between 2:00PM - 8:00PM EST an almost complete
computation of 300,000 digits of the MRB Constant was destroyed.
Computed for a long 4015. Hours (23.899 weeks or 1.4454*10^7
seconds) on a 2.66GH Core2Duo using 64 bit Windows XP. Max
memory used was 91 GB of RAM. The Mathematica 6.0 code used
follows:

Block[{$MaxExtraPrecision = 300000 + 8, a, b = -1, c = -1 - d,
d = (3 + Sqrt[8])^n, n = 131 Ceiling[300000/100], s = 0}, a[0] = 1;
d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++];
For[k = 0, k < n, c = b - c;
b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++];
N[1/2 - s/d, 300000]]
On September 18, 2008 a computation of 225,000 digits of MRB
Constant was started with a 2.66GH Core2Duo using 64 bit
Windows XP. It was completed in 1072 hours. Memory usage is
recorded in the attachment pt 225000.xls, near the bottom of this
post.

250,000 digits was attempted but failed to be completed to a
serious internal error which restarted the machine. The error
occurred sometime on December 24, 2008 between 9:00 AM and 9:
00 PM. The computation began on November 16, 2008 at 10:03
PM EST. Like the 300,000 digit computation this one was almost
complete when it failed. The Max memory used was 60.5 GB.

On Jan 29, 2009, 1:26:19 pm (UTC-0500) EST, I finished
computing 250,000 digits of the MRB constant. with a multiple step
Mathematica command running on a dedicated 64bit XP using
4Gb DDR2 Ram on board and 36 GB virtual. The computation took
only 333.102 hours. The digits are at http://marvinrayburns.
com/250KMRB.txt . The computation is completely documented in
the attached 250000.pd at bottom of this post.

On Sun 28 Mar 2010 21:44:50 (UTC-0500) EST, I started a
computation of 300000 digits of the MRB constant using an i7 with
8.0 GB of DDR3 Ram on board, but it failed due to hardware
problems.

I computed 299,998 Digits of the MRB constant. The computation
began Fri 13 Aug 2010 10:16:20 pm EDT and ended 2.23199*10^6
seconds later | Wednesday, September 8, 2010. I used
Mathematica 6.0 for Microsoft Windows (64-bit) (June 19, 2007)
That is an average of 7.44 seconds per digit.. I used my Dell Studio
XPS 8100 i7 860 @ 2.80 GH 2.80 GH with 8GB physical DDR3
RAM. Windows 7 reserved an additional 48.929 GB virtual Ram.

I computed exactly 300,000 digits to the right of the decimal point
of the MRB constant from Sat 8 Oct 2011 23:50:40 to Sat 5 Nov
2011 19:53:42 (2.405*10^6 seconds later). This run was 0.5766
seconds per digit slower than the 299,998 digit computation even
though it used 16GB physical DDR3 RAM on the same machine.
The working precision and accuracy goal combination were
maximized for exactly 300,000 digits, and the result was
automatically saved as a file instead of just being displayed on the
front end. Windows reserved a total of 63 GB of working memory of
which at 52 GB were recorded being used. The 300,000 digits
came from the Mathematica 7.0 command

Quit; DateString[]
digits = 300000; str = OpenWrite[]; SetOptions[str,
PageWidth -> 1000]; time = SessionTime[]; Write[str,
NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]},
WorkingPrecision -> digits + 3, AccuracyGoal -> digits,
Method -> "AlternatingSigns"]]; timeused =
SessionTime[] - time; here = Close[str]
DateString[]
314159 digits of the constant took 3 tries do to hardware failure.
Finishing on September 18, 2012 I computed 314159 digits, taking
59 GB of RAM. The digits are came from the Mathematica 8.0.4
code

DateString[]
NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]},
WorkingPrecision -> 314169, Method -> "AlternatingSigns"] //
Timing
DateString[]
Sam Noble of Apple computed 1,000,000 digits of the MRB
constant in 18 days 9 hours 11 minutes 34.253417 seconds.

Finishing on Dec 11, 2012 Ricard Crandall, an Apple scientist,
computed 1,048,576 digits in a lighting fast 76.4 hours (probably
processor time). That's on a 2.93 Ghz 8-core Nehalem. It took until
the use of DDR4 to compute nearly that many digits in an absolute
time that quick!!: In Aug of 2018 I computed 1,004,993 digits of the
MRB constant in 53.5 hours with 10 processor cores! Search this
post for "53.5" for documentation. Sept 21, 2018, I just now
computed 1,004,993 digits of the MRB constant in 50.37 hours of
absolute time (35.4 hours processor time) with 18 processor cores!
Search this post for "50.37 hours" for documentation.**

Previously, I computed a little over 1,200,000 digits of the MRB
constant in 11 days, 21 hours, 17 minutes, and 41 seconds,(
finishing on on March 31 2013). I used a six core Intel(R) Core(TM)
i7-3930K CPU @ 3.20 GHz 3.20 GHz.

On May 17, 2013 I finished a 2,000,000 or more digit computation
of the MRB constant, using only around 10GB of RAM. It took 37
days 5 hours 6 minutes 47.1870579 seconds. I used my six core
Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz 3.20 GHz.

A previous world record computation of the MRB constant was
finished on Sun 21 Sep 2014 18:35:06. It took 1 month 27 days 2
hours 45 minutes 15 seconds.The processor time from the
3,000,000+ digit computation was 22 days. I computed the
3,014,991 digits of the MRB constant with Mathematica 10.0. I
Used my new version of Richard Crandall's code in the attached
3M.nb, optimized for my platform and large computations. I also
used a six core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz 3.20
GHz with 64 GB of RAM of which only 16 GB was used. Can you
beat it (in more number of digits, less memory used, or less time
taken)? This confirms that my previous "2,000,000 or more digit
computation" was actually accurate to 2,009,993 digits. they were
used to check the first several digits of this computation. See
attached 3M.nb for the full code and digits.

Finished on Wed 16 Jan 2019 19:55:20, I computed over 4 million
digits of the MRB constant. It took 4 years of continuous tries. This
successful run took 65.13 days computation time, with a processor
time of 25.17 days, on a 3.7 GH overclocked up to 4.7 GH on all
cores Intel 6 core computer with 3000 MHz RAM. According to this
computation, the previous record, 3,000,000+ digit computation,
was accurate to 3,014,871 decimals, as this computation used my
own algorithm for computing n^(1/n) as found at chapter 3 in the
paper at

https://www.sciencedirect.
com/science/article/pii/0898122189900242

and the 3 million+ computation used Crandall's algorithm. Both
algorithms outperform Newton's method per calculation and
iteration.

See attached notebook.
MRB constant
Mouse over formula for proof