Triggering .click function with enter

I'm trying to make some kind of form. User have to click in order to send data further, but when user press 'enter' it returns error.

Script:

$('a.rec_submit').click(function(){
    form_update_mail(1);
    $('input.rec_submit_actual').click();
    return false;
}

HTML:

<a      class="rec_submit">SEND</a>
<input  class="rec_submit_actual" type="submit" />

I've searched a bit, and found that i can use keyup function in order for 'enter' to work also.

example:

$('a.rec_submit').on('keyup click', function(e){
    if (e.which === 13 || e.type === 'click') {
    form_update_mail(1);
    $('input.rec_submit_actual').on('keyup click', function();
    return false;
)}
});

Yet this doesnt work. Could someone explain what i'm doing wrong?

///// EDIT

None of the answers do work.

Form doesn't want to pass email further with 'enter', with 'click' it works just fine.

Error goes: email addres is not valid

Full script:

/*function form_update_mail()
{
var tresc_maila='';
$('#body_mail').val(tresc_maila);
};*/
var mail_string = '';

function form_update_mail(x) {
mail_string += $('input#component_recommend_to').val() + ',';
if (x == 1) mail_string = mail_string.substring(0, mail_string.length - 1);
$('#component_recommend_to_actual').val(mail_string);
}
$(document).ready(function() {
$('a#suggestproduct').click(function() {
    $(this).parent().fadeOut(500);
});
$('div.projektor_left_frame a, a.u10_polec').click(function() {
    $('div#component_projector_suggestproduct2').fadeIn(500);
    return false;
});
$('#component_recommend input').click(function() {
    if ($(this).attr('value') == 'wpisz e-mail' || $(this).attr('value') == 'wpisz imię') $(this).attr('value', '');
});
$('a.recommend_add_next').click(function() {
    form_update_mail(0);
    $('input#component_recommend_to').attr('value', '');
    $('input#component_recommend_name_to').attr('value', '');
    return false;
});
$('a.rec_submit').keypress(function(e) {
    if (e.which == 13 || e.which == 1) {
        form_update_mail(1);
        $('input.rec_submit_actual').click();
        return false;
    }
});
$('#component_recommend_body').val(tresc_maila);
tresc_maila = 'x';
$('#component_recommend_body').val(tresc_maila);
if ($('div.sender_email').length) $('#component_recommend_email').val($('div.sender_email').text());
if ($('div.sender_name').length) $('#component_recommend_name').val($('div.sender_name').text());
});

HTML:

<div id="component_recommend">
    <form action="contact.php" method="post">
        <input name="suggest_shop" type="hidden" value="submit">
        <div class="rec_main">
            <div class="rec_sub">
                <div style="margin-top: 30%; margin-bottom: 2%; font-size: 1.4em;">
                    Data 1:
                </div>
                <input  id="component_recommend_name"   name="firstname"    type="text" placeholder="Your name" class="rec_form" /><br />
                <input  id="component_recommend_email"  name="email"        type="text" placeholder="Your e-mail" style="margin-top: 5%" class="rec_form" />
                <div    style="margin-top: 8%; margin-bottom: 2%; font-size: 1.4em;">
                    Data 2:
                </div>
                <input      id="component_recommend_name_to"    name="name_to"  type="text" placeholder="His name" class="rec_form" /><br />
                <input      id="component_recommend_to"                         type="text" placeholder="His email" style="margin-top: 5%" class="rec_form" />
                <input      id="component_recommend_to_actual"  name="to"       type="text" placeholder="His email" style="display: none;" class="rec_form" />
                <textarea   id="component_recommend_body"       name="body"></textarea>
                <a          class="rec_submit">SEND</a>
                <input      class="rec_submit_actual" type="submit" />
            </div>
        </div>
    </form>
</div>