This is not exactly golf, and works only for nonnegative integers.

#!perl

use warnings;
use strict;

{
my $adder = '
        G
        s/^ *\([0-9]*\) *\n\([0-9]*\)$/0\1pm0\2nb0/;td;d;
        :d;s/n\(\(b\).*0\|\)[^0]*$/nb98765432109876543210a0123456789\2
+/;
        s/\(.\)\(p.*\)\(.\)\(n.*\1\).*a.*\3/\2\4\3/;
        s/^0*\(.*\)p\(.*m\)0*\(.*\)\(.\).\{11\}$/0\1p\4\20\3/;
        /m0*n.*0/!bd;
        s/^\(.*\)p\(.*\)m.*/\1\2/;s/^0*//;
        h;$!d
';
$adder =~ s/;/\n/g; # some seds don't like semicolons
sub printsum { 
        open my $SED, "|-", "sed", $adder or die "error running sed: $
+!";
        print $SED int($_), "\n" or die "error writing to sed: $!" 
                for @_;
        close $SED or die "error closing sed: $!";
}
}

my @array = (1,3,5,6);
printsum(@array);

__END__
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Update 2007-12-06: see Re^2: --- adding a column of integers

Update 2011-10-29: see also (almost) foldl.