how to randomly list 5 authors with at least 3 published posts

Question

Bit of a newbie here...

As the title states, I am trying to generate a list of 5 random users (taken from authors, editors and administrators) who have published at least 3 posts each. (Also, I would like to exclude the author whose user ID is 4.)

I think I have managed to do all this (hacking together various solutions gratefully taken from useful previous posts here in wordpress.stackexchange!) but I am not sure if this is the best solution or if it's coded correctly. I use this feature on many pages throughout my website and am worried it might be too db query intensive.

<?php
global $wpdb;
$min_posts = 3; 
$user_ids = $wpdb->get_col("
SELECT `post_author` 
FROM
  (SELECT `post_author`, 
    COUNT(*) AS `count` 
    FROM {$wpdb->posts}
    WHERE `post_status`='publish'  
    GROUP BY `post_author`) 
AS `stats`
WHERE `count` >= {$min_posts} AND post_author !=4;");
shuffle($user_ids);

for($someNumber = 1; $someNumber <= 5; $someNumber++) {
$user = get_userdata($user_ids[$someNumber]); ?>

      <?php echo get_avatar( $user->ID, '60' );?>
      <h7><a href="<?php echo get_author_posts_url( $user->ID ); ?>">
        <?php echo $user->display_name ;?></a></h7>
      <hr>
<?php
}
unset($user_ids);
?>

Any help truly appreciated, many thanks!