my @n = qw(16 107 108 110 112 114 115 117 118 119 120 121 122 123 124 
+125 127 128 130 132 133 135 136 138 142 146 149 150 154 156 157 158 1
+59 161 162);

my $set = Set::IntSpan::Fast->new;
$set->add(@n);
print $set->as_string, "\n";

__END__
16,107-108,110,112,114-115,117-125,127-128,130,132-133,135-136,138,142
+,146,149-150,154,156-159,161-162
[download]

How would you do it using paper and pencil, without a computer?

Please post your answer

Perhaps something like this?

#!/usr/bin/perl -w
use strict;

my @nums=qw(16 107 108 110 112 114 115 117 118 119 120 121 122 123 124
+ 125 127 128 130 132 133 135 136 138 142 146 149 150 154 156 157 158 
+159 161 162);

my $limit = 4;
my ($prev, $top) = shift @nums;

for (@nums){
    if ($prev + 1 == $_){
        if (++$top == $limit){
            die "Sequence of $limit found between @{[$_-$limit]} and $
+_\n";
        }
    }
    else {
        $top = 0;
    }
    $prev = $_;
}
[download]

-- 
Education is not the filling of a pail, but the lighting of a fire.
 -- W. B. Yeats

my @nums=qw(16 107 108 110 112 114 115 117 118 119 120 121 122 123 124
+ 125 127 128 130 132 133 135 136 138 142 146 149 150 154 156 157 158 
+159 161 162);

my $limit=4;

my $direction=1;

my @cont=get_cont($limit,$direction,@nums);

if(@cont>0)
    {
    print "First continuous block of $limit numbers in ",$direction>0?
+"ascending":"descending"," direction is: ",join(',',@cont);
    }
else
    {
    print "No block was found.\n";
    }

sub get_cont
{

my $limit=shift;

my $dir=shift;

my @nums=@_;

my $prev;
my @cont;

for $num (@nums)
    {
    
    if(($num-$prev)==$dir)
        {
        push(@cont,$num);
        
        if(@cont>=$limit)
            {
            return @cont;
            }
        }
    else
        {
        @cont=();
        }
        
    $prev=$num;
    
    }
    
return ();
}
[download]

i tried doing this but i get an error

 #!/usr/bin/perl -w
use strict;
use warnings;

    # program identifies if the number series contains > 2500 continuo
+us numbers
    
    # read this number  series into a arr1
    # if (length < 2500)
    #     then ,discrad
    # else {
    #     for (i=1 to $len)
    #         idenitfy the indices where there are breaks and
    #         store these indices into arr2

    #    foreach element in this @arr2
    #        if the difference in the indices > 2500
    #            then take the diff and print the range


    use Data::Dumper;
my @files = </Users/distance/8A/*>;
foreach my $file (@files){

    my $file = $ARGV[0];
    
    open INFILE," $file" or die "$!\n";

    <INFILE>;
    <INFILE>;
    my @arr1 = <INFILE>;
    close INFILE;

    my $len = scalar(@arr1) ;
    print "$len\n" ;
    
    my (@range, %h, @numbers);
    
    if ($len < 2500) {
        print "no use of searching\n" ;
    }
    else {
        my $last_line = $#arr1;
        my ($range_start, $range_end) = ('', '');

        for (my $i = $last_line; $i > 0; $i--)
        {
            my $line = $arr1[$i];

            if ($line =~ /t:\s*(\d+)/)
            {
                push(@numbers, $1);
            }
        }
    }
    
if (scalar @numbers){
    for (my $i = 0; $i< scalar(@numbers); $i++){
        last if ($i+1 > $#numbers);
        my $f = $numbers[$i];
        my $s = $numbers[$i+1];
        
        if ($f - $s == 1)
        {
            $h{s} = $f if (!$h{s}); 
            $h{e} = $s;
        }else{
            push(@range, [[$h{e}, $h{s}]]);
            $h{s} = 0;
            $h{e} = 0;
            
            # can be updated to any number to get all the ranges
            # for now gives only two ranges
            last if (scalar @range  == 2);
        }
        
    }
}
# in case its only one range

if ($h{s} && $h{e} && !scalar(@range))
{
    push (@range, [$h{e}, $h{s}]);
}

print Dumper \@range;
}
[download]

i get an error as= Use of uninitialized value $file in concatenation (.) or string at test.pl line 31

"Use of uninitialized value" is a warning, not an error. Line 31 of your posted program is

close INFILE;
[download]

so you are probably running a different version of the code.
Another suspicious thing is your open: do you really prepend a space to the argument?

Can you explain this portion?

     <INFILE>;
    <INFILE>;
    my @arr1 = <INFILE>;
    close INFILE;
[download]

From the context, I'm assuming your files have two header lines and then one number per line.

use strict;
use warnings;

for (rangeOver(*DATA, 4)) {
    print "$_->[0] to $_->[1] = " .
          ($_->[1] - $_->[0] + 1) . "\n";
}

sub rangeOver {
    my ($fh, $min) = @_;
    my ($start, $end, @ranges, $i);

    <$fh>; <$fh>;
    chomp(@_ = <$fh>);
    
    return [] if $#_ < $min - 1;
    
    $start = $end = shift;
    for (@_) {
        if ($_ == $end + 1) {
            $end++; next;
        }
        push @ranges, [$start, $end]
            if $end >= $start + $min - 1;
        $start = $end = $_;
    }
    push @ranges, [$start, $end]
        if $end >= $start + $min - 1;
    return @ranges;
}

__DATA__
some
header
1
3
4
5
6
8
10
11
12
13
14
[download]