If you do a print $num & $shift ? '+' : '.' ; in the loop (following diotalevi's example), you can observer for $num (0..20) the pattern:

........................
+.......................
.+......................
++......................
..+.....................
+.+.....................
.++.....................
+++.....................
...+....................
+..+....................
.+.+....................
++.+....................
..++....................
+.++....................
.+++....................
++++....................
....+...................
+...+...................
.+..+...................
++..+...................
..+.+...................

And the middle part (8388608..8388628):

.......................+
+......................+
.+.....................+
++.....................+
..+....................+
+.+....................+
.++....................+
+++....................+
...+...................+
+..+...................+
.+.+...................+
++.+...................+
..++...................+
+.++...................+
.+++...................+
++++...................+
....+..................+
+...+..................+
.+..+..................+
++..+..................+
..+.+..................+

And the end part (16777196..16777216):

..++.+++++++++++++++++++
+.++.+++++++++++++++++++
.+++.+++++++++++++++++++
++++.+++++++++++++++++++
....++++++++++++++++++++
+...++++++++++++++++++++
.+..++++++++++++++++++++
++..++++++++++++++++++++
..+.++++++++++++++++++++
+.+.++++++++++++++++++++
.++.++++++++++++++++++++
+++.++++++++++++++++++++
...+++++++++++++++++++++
+..+++++++++++++++++++++
.+.+++++++++++++++++++++
++.+++++++++++++++++++++
..++++++++++++++++++++++
+.++++++++++++++++++++++
.+++++++++++++++++++++++
++++++++++++++++++++++++
........................

Looks like a sparse matrix we could explore. Like, if you add if ($num<$shift){$total-=(24-$i)*(25+$i)/2; last;} in the loop, it would speed up the beginning of the loop (but slow down the end). Here a sample code (mostly diotalevi's):

#!/usr/bin/perl -w
use strict;
use integer;
my $ans = 0;
for my $num (0..16777216) {
  my $total = 0;
  my $shift = 1;
  for my $i (1..24) {
    $total += $num & $shift ?  $i : -$i; 
    if ($num<$shift){$total-=(24-$i)*(25+$i)/2; last;}
    $shift <<= 1;
  }
  $ans++ if $total == 0;
}
print $ans;
[download]

Ironically, I mostly found out how to slow the code down rather than speeding it up.

Though the $shift <<= 1; part is essentially static and equivalent to @shift = qw/ 1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 /;, using array will slow things down.

If you want something even slower, here is something I tried in Pari:

#!/usr/bin/perl -w
use Math::Pari qw/ PARI / ;
PARI 'ans=0' ;
my $cmd = qq/ 
            for(
                num = 0, 1677,
                total=0; shft=1;
                for(
                    i = 1, 24,
                    if(bitand(num,shft)==0, total+=-i, total+=i);
                    shft <<= 1;
                );
                if(total==0, ans++);
            ) 
          /;
$cmd =~ s/\s//g;
PARI $cmd ;
print PARI('ans');
[download]

Funny enough, it's slow as CDROM.

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