The distributions of the integers of a continued fraction given by Gauss-Kuzmin is different from I had worked out some years ago. They give P(an = k) = −ln(1 − (k + 1)−2). I got the simpler result P(an = k) = 6/(πk)2

Guess what! Gauss was right. This program contrasts my distribution with Gauss’:

```#include <stdio.h>
#include <math.h>
typedef double f;
static f sq(f x){return x*x;}
static f lg(f x){return log(x)/log(2);}
int main(){f ms=0, gs=0;
f pi = 3.141592653589793238;
f nrm = pi*pi/6;
int j;
for(j = 1; j<20000; ++j) {f m = 1/(sq(j)*nrm), g = -lg(1-1/sq(j+1));
ms += m; gs += g;
if(j<21) printf("%d %19.16f %19.16f\n", j, m, g);}
printf("%19.16f %19.16f\n", ms, gs);
return 0;}
```
which yields:
```1  0.6079271018540267  0.4150374992788438
2  0.1519817754635067  0.1699250014423125
3  0.0675474557615585  0.0931094043914815
4  0.0379954438658767  0.0588936890535686
5  0.0243170840741611  0.0406419844973459
6  0.0168868639403896  0.0297473433940521
7  0.0124066755480414  0.0227200765000835
8  0.0094988609664692  0.0179219079972625
9  0.0075052728623954  0.0144995696951151
10  0.0060792710185403  0.0119726416660759
11  0.0050241909244134  0.0100536646639229
12  0.0042217159850974  0.0085620135034241
13  0.0035972017861185  0.0073795303655975
14  0.0031016688870103  0.0064262691594330
15  0.0027018982304623  0.0056465631411421
16  0.0023747152416173  0.0050006810583665
17  0.0021035539856541  0.0044596481906998
18  0.0018763182155988  0.0040019305574962
19  0.0016840085923934  0.0036112535523788
20  0.0015198177546351  0.0032751320328610
0.9999696028849843  0.9999278670512570
```
and this Algol68 program, using 4000 digits of precision:
```INT lm = 100; INT big := 0;
MODE B = LONG LONG REAL;
[lm] INT h;
FOR j TO lm DO h[j] := 0 OD;
B x := long long pi;
TO 10000 DO INT q = SHORTEN SHORTEN ENTIER x;
IF q <= lm THEN h[q] +:= 1 ELSE big +:= 1 FI;
x := 1/(x-q) OD;
FOR j TO lm DO print((j, h[j], newline)) OD;
print(big)
```
considers the first 10000 terms for π and reports
```         +1      +4182
+2      +1675
+3       +939
+4       +595
+5       +404
+6       +286
+7       +254
+8       +181
+9       +123
+10       +129
+11        +77
+12        +88
+13        +76
+14        +71
+15        +58
+16        +46
+17        +43
+18        +43
+19        +30
+20        +31
+21        +26
+22        +22
+23        +30
+24        +28
+25        +17
+26        +18
+27        +14
+28        +11
+29        +17
+30        +17
+31        +12
+32        +20
+33        +14
+34        +10
+35         +8
+36        +14
+37        +12
+38         +4
+39         +8
+40         +9
+41        +10
+42         +9
+43         +9
+44         +5
+45         +6
+46         +5
+47        +11
+48         +7
+49         +3
+50        +11
+51         +3
+52         +6
+53        +11
+54         +5
+55        +10
+56         +3
+57         +6
+58         +2
+59         +6
+60         +5
+61         +4
+62         +4
+63         +6
+64         +1
+65         +4
+66         +4
+67         +4
+68         +2
+69         +1
+70         +3
+71         +1
+72         +3
+73         +7
+74         +1
+75         +0
+76         +0
+77         +2
+78         +2
+79         +2
+80         +5
+81         +0
+82         +0
+83         +0
+84         +6
+85         +1
+86         +2
+87         +1
+88         +4
+89         +0
+90         +0
+91         +1
+92         +2
+93         +2
+94         +4
+95         +3
+96         +0
+97         +1
+98         +1
+99         +2
+100         +0
+139
```