esri image server download.txt

I used (converted from Python):
        public static (int X, int Y) Deg2Tile(double latDeg, double lonDeg, int zoom)
        {
            var latRad = (latDeg * Math.PI) / 180;
            var n = Math.Pow(2.0, zoom);
            var xtile = (int)((lonDeg + 180.0) / 360.0 * n);
            var ytile = (int)((1.0 - Math.Log(Math.Tan(latRad) + (1 / Math.Cos(latRad))) / Math.PI) / 2.0 * n);
            return (xtile, ytile);
        }

From:
https://gis.stackexchange.com/questions/190966/export-arcgis-tiles-data-to-any-image-format
  https://wiki.openstreetmap.org/wiki/Slippy_map_tilenames

To convert between lat/long and tile number using C#:

    int long2tilex(double lon, int z)
    {
        return (int)(Math.Floor((lon + 180.0) / 360.0 * (1 << z)));
    }

    int lat2tiley(double lat, int z)
    {
        return (int)Math.Floor((1 - Math.Log(Math.Tan(ToRadians(lat)) + 1 / Math.Cos(ToRadians(lat))) / Math.PI) / 2 * (1 << z));
    }

    double tilex2long(int x, int z)
    {
        return x / (double)(1 << z) * 360.0 - 180;
    }

    double tiley2lat(int y, int z)
    {
        double n = Math.PI - 2.0 * Math.PI * y / (double)(1 << z);
        return 180.0 / Math.PI * Math.Atan(0.5 * (Math.Exp(n) - Math.Exp(-n)));
    }
    
Alternatively:
    public PointF WorldToTilePos(double lon, double lat, int zoom)
    {
      PointF p = new Point();
      p.X = (float)((lon + 180.0) / 360.0 * (1 << zoom));
      p.Y = (float)((1.0 - Math.Log(Math.Tan(lat * Math.PI / 180.0) + 
        1.0 / Math.Cos(lat * Math.PI / 180.0)) / Math.PI) / 2.0 * (1 << zoom));

      return p;
    }

    public PointF TileToWorldPos(double tile_x, double tile_y, int zoom) 
    {
      PointF p = new Point();
      double n = Math.PI - ((2.0 * Math.PI * tile_y) / Math.Pow(2.0, zoom));

      p.X = (float)((tile_x / Math.Pow(2.0, zoom) * 360.0) - 180.0);
      p.Y = (float)(180.0 / Math.PI * Math.Atan(Math.Sinh(n)));

      return p;
    }