Merge pull request #6297 from drbaggy/master

Some notes - and numbers.... 

5 files changed, 149 insertions, 69 deletions

diff --git a/challenge-169/james-smith/README.md b/challenge-169/james-smith/README.md
index ba9c64a2b4..5c699ca4f3 100644
--- a/challenge-169/james-smith/README.md
+++ b/challenge-169/james-smith/README.md
@@ -30,7 +30,7 @@ For flexibility we define the max count `$MAX` as the command-line argument if o
 
 ```perl
 for( my( $MAX, $c, $n, @f ) = ($ARGV[0] // 1e2, 0); $c<$MAX; ) {
-  printf "%8d: %10d = %5d x %d\n", ++$c, $n, @f if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1];
+  printf "%8d: %10d = %5d x %d\n", ++$c, $n, @f if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1]
 }
 ```
 
@@ -44,6 +44,8 @@ The output in each row is the brilliant number and the two primes which are it's
 
 **Moan:** Why is there no `sayf` function similar to `printf` - using `say sprintf` is just "messy" each time...
 
+The first 50 brilliant numbers are:
+
 ```
        1:          4 =     2 x 2
        2:          6 =     2 x 3
@@ -96,14 +98,13 @@ The output in each row is the brilliant number and the two primes which are it's
       49:        703 =    19 x 37
       50:        713 =    23 x 31
 ```
-
 ### Removing pretty print
 
 If we remove the pretty print this reduces to:
 
 ```perl
 for( my( $c, $n, @f ) = 100; $c; ) {
-  $c--, say $n if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1];
+  $c--, say $n if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1]
 }
 ```
 
@@ -133,62 +134,141 @@ We then compute the `gcd` of these powers - if it is 1 then we display the resul
 ```perl
 for( my( $MAX, $c, $n, @f ) = ($ARGV[0] // 1e2, 0); $c<$MAX; ) {
   printf "%6d: %15d = %s\n", ++$c, $n, join ' . ', map  { join '^', @$_ } @f
-    if 1 == gcd map { $_->[1] < 2 ? next : $_->[1] } @f = factor_exp ++$n;
+    if 1 == gcd map { $_->[1] < 2 ? next : $_->[1] } @f = factor_exp ++$n
 }
 ```
 
 The following are the first 50 achilles numbers.
-```
-     1:              72 = 2^3 . 3^2
-     2:             108 = 2^2 . 3^3
-     3:             200 = 2^3 . 5^2
-     4:             288 = 2^5 . 3^2
-     5:             392 = 2^3 . 7^2
-     6:             432 = 2^4 . 3^3
-     7:             500 = 2^2 . 5^3
-     8:             648 = 2^3 . 3^4
-     9:             675 = 3^3 . 5^2
-    10:             800 = 2^5 . 5^2
-    11:             864 = 2^5 . 3^3
-    12:             968 = 2^3 . 11^2
-    13:             972 = 2^2 . 3^5
-    14:            1125 = 3^2 . 5^3
-    15:            1152 = 2^7 . 3^2
-    16:            1323 = 3^3 . 7^2
-    17:            1352 = 2^3 . 13^2
-    18:            1372 = 2^2 . 7^3
-    19:            1568 = 2^5 . 7^2
-    20:            1800 = 2^3 . 3^2 . 5^2
-    21:            1944 = 2^3 . 3^5
-    22:            2000 = 2^4 . 5^3
-    23:            2312 = 2^3 . 17^2
-    24:            2592 = 2^5 . 3^4
-    25:            2700 = 2^2 . 3^3 . 5^2
-    26:            2888 = 2^3 . 19^2
-    27:            3087 = 3^2 . 7^3
-    28:            3200 = 2^7 . 5^2
-    29:            3267 = 3^3 . 11^2
-    30:            3456 = 2^7 . 3^3
-    31:            3528 = 2^3 . 3^2 . 7^2
-    32:            3872 = 2^5 . 11^2
-    33:            3888 = 2^4 . 3^5
-    34:            4000 = 2^5 . 5^3
-    35:            4232 = 2^3 . 23^2
-    36:            4500 = 2^2 . 3^2 . 5^3
-    37:            4563 = 3^3 . 13^2
-    38:            4608 = 2^9 . 3^2
-    39:            5000 = 2^3 . 5^4
-    40:            5292 = 2^2 . 3^3 . 7^2
-    41:            5324 = 2^2 . 11^3
-    42:            5400 = 2^3 . 3^3 . 5^2
-    43:            5408 = 2^5 . 13^2
-    44:            5488 = 2^4 . 7^3
-    45:            6075 = 3^5 . 5^2
-    46:            6125 = 5^3 . 7^2
-    47:            6272 = 2^7 . 7^2
-    48:            6728 = 2^3 . 29^2
-    49:            6912 = 2^8 . 3^3
-    50:            7200 = 2^5 . 3^2 . 5^2
+
+| Index        | Value                | Factors                                                      |
+| -----------: | -------------------: | ------------------------------------------------------------ |
+|            1 |                   72 | 2<sup>3</sup>.3<sup>2</sup>                                  |
+|            2 |                  108 | 2<sup>2</sup>.3<sup>3</sup>                                  |
+|            3 |                  200 | 2<sup>3</sup>.5<sup>2</sup>                                  |
+|            4 |                  288 | 2<sup>5</sup>.3<sup>2</sup>                                  |
+|            5 |                  392 | 2<sup>3</sup>.7<sup>2</sup>                                  |
+|            6 |                  432 | 2<sup>4</sup>.3<sup>3</sup>                                  |
+|            7 |                  500 | 2<sup>2</sup>.5<sup>3</sup>                                  |
+|            8 |                  648 | 2<sup>3</sup>.3<sup>4</sup>                                  |
+|            9 |                  675 | 3<sup>3</sup>.5<sup>2</sup>                                  |
+|           10 |                  800 | 2<sup>5</sup>.5<sup>2</sup>                                  |
+|           11 |                  864 | 2<sup>5</sup>.3<sup>3</sup>                                  |
+|           12 |                  968 | 2<sup>3</sup>.11<sup>2</sup>                                 |
+|           13 |                  972 | 2<sup>2</sup>.3<sup>5</sup>                                  |
+|           14 |                1,125 | 3<sup>2</sup>.5<sup>3</sup>                                  |
+|           15 |                1,152 | 2<sup>7</sup>.3<sup>2</sup>                                  |
+|           16 |                1,323 | 3<sup>3</sup>.7<sup>2</sup>                                  |
+|           17 |                1,352 | 2<sup>3</sup>.13<sup>2</sup>                                 |
+|           18 |                1,372 | 2<sup>2</sup>.7<sup>3</sup>                                  |
+|           19 |                1,568 | 2<sup>5</sup>.7<sup>2</sup>                                  |
+|           20 |                1,800 | 2<sup>3</sup>.3<sup>2</sup>.5<sup>2</sup>                    |
+|           21 |                1,944 | 2<sup>3</sup>.3<sup>5</sup>                                  |
+|           22 |                2,000 | 2<sup>4</sup>.5<sup>3</sup>                                  |
+|           23 |                2,312 | 2<sup>3</sup>.17<sup>2</sup>                                 |
+|           24 |                2,592 | 2<sup>5</sup>.3<sup>4</sup>                                  |
+|           25 |                2,700 | 2<sup>2</sup>.3<sup>3</sup>.5<sup>2</sup>                    |
+|           26 |                2,888 | 2<sup>3</sup>.19<sup>2</sup>                                 |
+|           27 |                3,087 | 3<sup>2</sup>.7<sup>3</sup>                                  |
+|           28 |                3,200 | 2<sup>7</sup>.5<sup>2</sup>                                  |
+|           29 |                3,267 | 3<sup>3</sup>.11<sup>2</sup>                                 |
+|           30 |                3,456 | 2<sup>7</sup>.3<sup>3</sup>                                  |
+|           31 |                3,528 | 2<sup>3</sup>.3<sup>2</sup>.7<sup>2</sup>                    |
+|           32 |                3,872 | 2<sup>5</sup>.11<sup>2</sup>                                 |
+|           33 |                3,888 | 2<sup>4</sup>.3<sup>5</sup>                                  |
+|           34 |                4,000 | 2<sup>5</sup>.5<sup>3</sup>                                  |
+|           35 |                4,232 | 2<sup>3</sup>.23<sup>2</sup>                                 |
+|           36 |                4,500 | 2<sup>2</sup>.3<sup>2</sup>.5<sup>3</sup>                    |
+|           37 |                4,563 | 3<sup>3</sup>.13<sup>2</sup>                                 |
+|           38 |                4,608 | 2<sup>9</sup>.3<sup>2</sup>                                  |
+|           39 |                5,000 | 2<sup>3</sup>.5<sup>4</sup>                                  |
+|           40 |                5,292 | 2<sup>2</sup>.3<sup>3</sup>.7<sup>2</sup>                    |
+|           41 |                5,324 | 2<sup>2</sup>.11<sup>3</sup>                                 |
+|           42 |                5,400 | 2<sup>3</sup>.3<sup>3</sup>.5<sup>2</sup>                    |
+|           43 |                5,408 | 2<sup>5</sup>.13<sup>2</sup>                                 |
+|           44 |                5,488 | 2<sup>4</sup>.7<sup>3</sup>                                  |
+|           45 |                6,075 | 3<sup>5</sup>.5<sup>2</sup>                                  |
+|           46 |                6,125 | 5<sup>3</sup>.7<sup>2</sup>                                  |
+|           47 |                6,272 | 2<sup>7</sup>.7<sup>2</sup>                                  |
+|           48 |                6,728 | 2<sup>3</sup>.29<sup>2</sup>                                 |
+|           49 |                6,912 | 2<sup>8</sup>.3<sup>3</sup>                                  |
+|           50 |                7,200 | 2<sup>5</sup>.3<sup>2</sup>.5<sup>2</sup>                    |
+
+
+Some examples for larger values of `n` are:
+
+| Index        | Value                | Factors                                                      |
+| -----------: | -------------------: | ------------------------------------------------------------ |
+|          100 |               21,600 | 2<sup>5</sup>.3<sup>3</sup>.5<sup>2</sup>                    |
+|          200 |               66,248 | 2<sup>3</sup>.7<sup>2</sup>.13<sup>2</sup>                   |
+|          300 |              136,107 | 3<sup>3</sup>.71<sup>2</sup>                                 |
+|          400 |              225,000 | 2<sup>3</sup>.3<sup>2</sup>.5<sup>5</sup>                    |
+|          500 |              333,396 | 2<sup>2</sup>.3<sup>5</sup>.7<sup>3</sup>                    |
+|          600 |              464,648 | 2<sup>3</sup>.241<sup>2</sup>                                |
+|          700 |              617,400 | 2<sup>3</sup>.3<sup>2</sup>.5<sup>2</sup>.7<sup>3</sup>      |
+|          800 |              784,000 | 2<sup>7</sup>.5<sup>3</sup>.7<sup>2</sup>                    |
+|          900 |              969,624 | 2<sup>3</sup>.3<sup>3</sup>.67<sup>2</sup>                   |
+|        1,000 |            1,179,648 | 2<sup>17</sup>.3<sup>2</sup>                                 |
+|        2,000 |            4,255,443 | 3<sup>3</sup>.397<sup>2</sup>                                |
+|        3,000 |            9,082,800 | 2<sup>4</sup>.3<sup>3</sup>.5<sup>2</sup>.29<sup>2</sup>     |
+|        4,000 |           15,635,232 | 2<sup>5</sup>.3<sup>2</sup>.233<sup>2</sup>                  |
+|        5,000 |           23,876,179 | 19<sup>3</sup>.59<sup>2</sup>                                |
+|        6,000 |           33,818,428 | 2<sup>2</sup>.7<sup>3</sup>.157<sup>2</sup>                  |
+|        7,000 |           45,489,708 | 2<sup>2</sup>.3<sup>3</sup>.11<sup>2</sup>.59<sup>2</sup>    |
+|        8,000 |           58,752,800 | 2<sup>5</sup>.5<sup>2</sup>.271<sup>2</sup>                  |
+|        9,000 |           73,641,248 | 2<sup>5</sup>.37<sup>2</sup>.41<sup>2</sup>                  |
+|       10,000 |           90,209,312 | 2<sup>5</sup>.23<sup>2</sup>.73<sup>2</sup>                  |
+|       20,000 |          344,478,752 | 2<sup>5</sup>.17<sup>2</sup>.193<sup>2</sup>                 |
+|       30,000 |          758,595,456 | 2<sup>7</sup>.3<sup>5</sup>.29<sup>3</sup>                   |
+|       40,000 |        1,330,259,301 | 3<sup>3</sup>.7<sup>3</sup>.379<sup>2</sup>                  |
+|       50,000 |        2,057,748,300 | 2<sup>2</sup>.3<sup>7</sup>.5<sup>2</sup>.97<sup>2</sup>     |
+|       60,000 |        2,941,077,600 | 2<sup>5</sup>.3<sup>7</sup>.5<sup>2</sup>.41<sup>2</sup>     |
+|       70,000 |        3,978,593,667 | 3<sup>3</sup>.61<sup>2</sup>.199<sup>2</sup>                 |
+|       80,000 |        5,171,352,984 | 2<sup>3</sup>.3<sup>5</sup>.7<sup>2</sup>.233<sup>2</sup>    |
+|       90,000 |        6,518,604,456 | 2<sup>3</sup>.3<sup>2</sup>.7<sup>2</sup>.13<sup>3</sup>.29<sup>2</sup> |
+|      100,000 |        8,017,975,944 | 2<sup>3</sup>.3<sup>3</sup>.11<sup>3</sup>.167<sup>2</sup>   |
+|      125,000 |       12,437,566,904 | 2<sup>3</sup>.7<sup>3</sup>.2129<sup>2</sup>                 |
+|      150,000 |       17,810,638,848 | 2<sup>11</sup>.3<sup>2</sup>.983<sup>2</sup>                 |
+|      175,000 |       24,140,196,992 | 2<sup>7</sup>.31<sup>2</sup>.443<sup>2</sup>                 |
+|      200,000 |       31,413,171,744 | 2<sup>5</sup>.3<sup>2</sup>.17<sup>3</sup>.149<sup>2</sup>   |
+|      225,000 |       39,636,156,125 | 5<sup>3</sup>.17807<sup>2</sup>                              |
+|      250,000 |       48,804,377,888 | 2<sup>5</sup>.7<sup>4</sup>.797<sup>2</sup>                  |
+|      275,000 |       58,919,206,088 | 2<sup>3</sup>.85819<sup>2</sup>                              |
+|      300,000 |       69,976,609,587 | 3<sup>3</sup>.50909<sup>2</sup>                              |
+|      325,000 |       81,981,196,443 | 3<sup>3</sup>.55103<sup>2</sup>                              |
+|      350,000 |       94,917,245,000 | 2<sup>3</sup>.5<sup>4</sup>.4357<sup>2</sup>                 |
+
+And these are the first numbers with `n` digits...
+
+| Index        | Value                | Factors                                                      |
+| -----------: | -------------------: | ------------------------------------------------------------ |
+|            1 |                   72 | 2<sup>3</sup>.3<sup>2</sup>                                  |
+|            2 |                  108 | 2<sup>2</sup>.3<sup>3</sup>                                  |
+|           14 |                1,125 | 3<sup>2</sup>.5<sup>3</sup>                                  |
+|           61 |               10,125 | 3<sup>4</sup>.5<sup>3</sup>                                  |
+|          253 |              100,352 | 2<sup>11</sup>.7<sup>2</sup>                                 |
+|          917 |            1,000,188 | 2<sup>2</sup>.3<sup>6</sup>.7<sup>3</sup>                    |
+|        3,159 |           10,011,125 | 5<sup>3</sup>.283<sup>2</sup>                                |
+|       10,554 |          100,018,800 | 2<sup>4</sup>.3<sup>6</sup>.5<sup>2</sup>.7<sup>3</sup>      |
+|       34,562 |        1,000,042,200 | 2<sup>3</sup>.3<sup>6</sup>.5<sup>2</sup>.19<sup>3</sup>     |
+|      111,892 |       10,000,373,888 | 2<sup>7</sup>.8839<sup>2</sup>                               |
+|      359,341 |      100,001,075,328 | 2<sup>7</sup>.3<sup>2</sup>.7<sup>2</sup>.11<sup>6</sup>     |
+
+### Github formatting script....
+
+To convert from the human readable text file generated by the script to github markup I wrote this script:
+
+```perl
+head();
+while( <> ) {
+  head(), next unless /\S/;
+  my ($c,$n,$p) = m{(\d+):\s+(\d+)\s+=\s+(.*)$};
+  printf "| %12s | %20s | %-60s |\n", th($c), th($n),
+    join q(.), map { sprintf '%d<sup>%d</sup>', split /\^/ } split / \. /, $p
+}
+
+sub head { print "\n| Index        | Value                | Factors                                                      |
+| -----------: | -------------------: | ------------------------------------------------------------ |\n" }
+sub th { scalar reverse ( (reverse $_[0]) =~ s/(\d\d\d)(?=\d)(?!\d*\.)/$1,/gr ) }
 ```
 ### Removing pretty print
 
@@ -196,7 +276,7 @@ If we remove the pretty print this reduces to:
 
 ```perl
 for( my( $c, $n ) = 100; $c; ) {
-  $c--, say $n if 1 == gcd map{ $_->[1] < 2 ? next : $_->[1] } factor_exp ++$n;
+  $c--, say $n if 1 == gcd map{ $_->[1] < 2 ? next : $_->[1] } factor_exp ++$n
 }
 ```
 
diff --git a/challenge-169/james-smith/perl/ch-1-npp.pl b/challenge-169/james-smith/perl/ch-1-npp.pl
index ce0b88c777..6b5c2d4c3b 100644
--- a/challenge-169/james-smith/perl/ch-1-npp.pl
+++ b/challenge-169/james-smith/perl/ch-1-npp.pl
@@ -10,9 +10,9 @@ use Time::HiRes qw(time);
 my $time = time;
 
 #-------------------------------------------------------------------------------
-for( my( $c, $n, @f ) = 100; $c; ) {
-  $c--, say $n if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1];
+for( my( $c, $n, @f ) = $ARGV[0] // 100; $c; ) {
+  $c--, say $n if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1]
 }
 
-warn 'Time taken: ', time-$time, "\n";
+warn 'Time taken: ', time-$time, "\n"
 
diff --git a/challenge-169/james-smith/perl/ch-1.pl b/challenge-169/james-smith/perl/ch-1.pl
index 775040f182..5671e8ce07 100644
--- a/challenge-169/james-smith/perl/ch-1.pl
+++ b/challenge-169/james-smith/perl/ch-1.pl
@@ -13,10 +13,10 @@ my $time = time;
 # must have preciesely 2 prime factors
 # THEN each factor must be the same length;
 
-for( my( $MAX, $c, $n, @f ) = ($ARGV[0] // 1e2,0); $c < $MAX; ) {
+for( my( $MAX, $c, $n, @f ) = ( $ARGV[0] // 100, 0 ); $c < $MAX; ) {
   printf "%8d: %10d = %5d x %d\n", ++$c, $n, @f
-    if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1];
+    if 2 == ( @f = factor ++$n ) && length $f[0] == length $f[1]
 }
 
-warn 'Time taken: ', time-$time, "\n";
+warn 'Time taken: ', time-$time, "\n"
 
diff --git a/challenge-169/james-smith/perl/ch-2-npp.pl b/challenge-169/james-smith/perl/ch-2-npp.pl
index a3d686dc25..590e015325 100644
--- a/challenge-169/james-smith/perl/ch-2-npp.pl
+++ b/challenge-169/james-smith/perl/ch-2-npp.pl
@@ -10,9 +10,9 @@ use Time::HiRes qw(time);
 my $time = time;
 
 #-------------------------------------------------------------------------------
-for( my( $c, $n ) = ( 100 ); $c; ) {
-  $c--, say $n if 1 == gcd map{ $_->[1] < 2 ? next : $_->[1] } factor_exp ++$n;
+for( my( $c, $n ) = $ARGV[0] // 100; $c; ) {
+  $c--, say $n if 1 == gcd map{ $_->[1] < 2 ? next : $_->[1] } factor_exp ++$n
 }
 
-warn 'Time taken: ', time-$time, "\n";
+warn 'Time taken: ', time-$time, "\n"
 
diff --git a/challenge-169/james-smith/perl/ch-2.pl b/challenge-169/james-smith/perl/ch-2.pl
index c55acdc58d..f6477524ae 100644
--- a/challenge-169/james-smith/perl/ch-2.pl
+++ b/challenge-169/james-smith/perl/ch-2.pl
@@ -1,4 +1,4 @@
-#!/usr/local/bin/perl
+#/usr/local/bin/perl
 
 use strict;
 
@@ -26,10 +26,10 @@ my $time = time;
 # To pretty print the archilles numbers - we use our counter, and display
 # it alongside the number and the factorisation.
 
-for( my( $MAX, $c, $n, @f ) = ($ARGV[0] // 1e2,0); $c<$MAX; ) {
+for( my( $MAX, $c, $n, @f ) = ( $ARGV[0] // 100, 0 ); $c < $MAX; ) {
   say sprintf '%6d: %15d = %s', ++$c, $n, join ' . ', map  { join '^', @$_ } @f
-    if 1 == gcd map { $_->[1] < 2 ? next : $_->[1] } @f = factor_exp ++$n;
+    if 1 == gcd map { $_->[1] < 2 ? next : $_->[1] } @f = factor_exp ++$n
 }
 
-warn 'Time taken: ', time-$time, "\n";
+warn 'Time taken: ', time-$time, "\n"