while (<>) {
   say if $. % 16 == 0;
}
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$., %

You forgot to reset line numbers across file boundaries.

FWIW, if reading from STDIN, there are no file boundaries :)

ikegami really proposed the easiest solution; file line number (represented in $.), modulus 16. If there's no "remainder", you're on a line that is a multiple of 16. You could, as an alternative for math-challenged individuals, increment a counter, and when it gets to the trigger value reset it and print the line. But that's more work just for the sake of being lazy. ;)

Oh, that reminds me of why I replied to this post (after an obviously perfect answer has already been given)... You are asking how to read every 16th line, but you really should be asking how to ignore all but every 16th line. Or in other words, read every line but only take action on multiples of 16. There is no easy way to only read every 16th line, if lines are of unknown length. That's a little like asking how to find "steam" in the dictionary without thumbing through it to find it.

There is the other lazy and inefficient alternative of using Tie::File to gain an index to each line, but then you still have to do math, so it really doesn't buy you anything other than the ability to easily access any line at will.

UPDATE:

more explicit:

while ( @lines = readlines($filehandle,16) ) {
  print @lines,"----\n";
}
 
sub readlines {
  my ($fh, $count) = @_;
  my @gulp;
  while (<$fh>) {
    push @gulp,$_;
    last unless --$count;
  }
  return @gulp;
}
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If I misunderstood and you are trying to get blocks of 16 lines,

my @buf;
while (<>) {
   push @buf, $_;
   if (@buf == 16) {
      do_it(\@buf);
      @buf = ();
   }
}

if (@buf) {
   do_it(\@buf);
}
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Update: but only if the rest of the logic is simple enough for inclusion in a one-liner like this.

update2: modified to shift target line nos to 16,32 etc. - a weird brainbug made me think 1,17 etc. was wanted - I need a holiday, lol!

my $line_number = 16;
my $count = 0;

while(<DATA>){
  chomp($_);
  $count++;
  next unless $count == $line_number;
  print "$_\n";
  $count = 0;
}

__DATA__
1 Bob 48 IT
2 David 26 IT
3 Jerry 22  IT
4 Josh 33  IT
5 Karl 35  IT
6 Dan 50  IT
7 Tom 26 IT
8 Randy 48  Engineering
9 Trevor 57 Engineering
10 Tony 48 Engineering
11 Larry 39 Engineering
12 Farhan 40  Engineering
13 Sandy 28 Engineering
14 Jessie 55 Engineering
15 Carl 25 Engineering
16 Donald 58 Engineering
17 Rick 44 Engineering
18 Barry 38 Engineering
19 Bruce 30 Engineering
20 Joe 37 Engineering
21 Mun 53 Engineering
22 Shiro 48 Engineering
23 Sally 50 HR
24 John 32 Finance
[download]