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JavaScript

var data = {
  "type": "FeatureCollection",
  "features": [{
    "type": "Feature",
    "properties": {
      "time": 1438342780,
      "title": "Iran's foreign minister calls for world's nuclear weapons states to disarm",
      "author": "Julian Borger",
      "web_id": "world/2015/jul/31/iran-nuclear-weapons-states-disarm-israel"
    },
    "geometry": {
      "type": "Point",
      "coordinates": [-77.26526,
        38.90122
      ]
    }
  }, {
    "type": "Feature",
    "properties": {
      "time": 1438300867,
      "title": "Big bangs over the white cliffs of Dover as unique 1915 artillery gun is fired again",
      "author": "Maev Kennedy",
      "web_id": "world/2015/jul/31/big-bangs-over-white-cliffs-dover-unique-1915-artillery-gun-fired-again"
    },
    "geometry": {
      "type": "Point",
      "coordinates": [
        1.3,
        51.13333
      ]
    }
  }]
}

function getDate(time) {
  var d = new Date(0);
  d.setUTCSeconds(time);
  return [d.getMonth() + 1, d.getDate(), d.getFullYear()].join('/');
}

function getData(data) {
  var obj = data.features.reduce(function(p, c) {
    var date = getDate(c.properties.time);
    p[date] = (p[date] + 1) || 1;
    return p;
  }, {});
  return Object.keys(obj).map(function (el) {
    return { date: el, number: obj[el] };
  })
}

var result = getData(data);
console.log(result)