OpenLayers, how compute the tile "name" under the mouse

I have working on a web page using OpenLayers which among others shows a 
OpenStreetMap layer.
The issue is I need to move the mouse over the map and print on a label the tile name 
in the form of z/x/y, for example, 5/15/10.

The approach to done this is:

1. register a listener to the OpenLayers map instance and then, for every mouse event triggered,
2. get the lonlat and translate it from current map projection to the EPSG:4326 (WGS84).

map.events.register("mousemove", this, function (evt) {
  var lonlat = new OpenLayers.LonLat(0, 0);
  if (evt) {
    lonlat = map.getLonLatFromPixel(evt.xy);
    lonlat.transform(
      this.map.getProjectionObject(),
      newOpenLayers.Projection("EPSG:4326")
    );
  }
  tileX = long2tile(lonlat.lon, map.getZoom());
  tileY = latg2tile(lonlat.lat, map.getZoom());
  tileZ = map.getZoom();

  // Print the tilez, tilex and tiley values
});

To compute the tile X,Y grid coordinate from latlon you can use next functions (obtained from OpenStreetMaps site):

// Functions to compute tiles from lat/lon
function long2tile(lon, zoom) {
  return Math.floor(((lon + 180) / 360) * Math.pow(2, zoom));
}
function lat2tile(lat, zoom) {
  return Math.floor(
    ((1 -
      Math.log(
        Math.tan((lat * Math.PI) / 180) + 1 / Math.cos((lat * Math.PI) / 180)
      ) /
        Math.PI) /
      2) *
      Math.pow(2, zoom)
  );
}