bases.scala

// This prints the top 10 alternative bases, ordered by the (# of factors, excluding 1 & base) / base.
// So, the last number printed has the highest number of factors relative to the size of the base.
//
// Turns out that 12 is a great base!
object Main extends App {
  // 7560 is the smallest number that is divisible by all number between 1 and 10.
  val facMap = (1 to 7560).foldLeft(Map.empty[List[Int], Int]) { (acc, n) =>
    val factors = (2 to (n / 2)).filter(n % _ == 0).toList
    if (acc contains factors) acc
    else acc + (factors -> n)
  }

  val result = facMap.toList.map { case (factors, n) =>
    (factors.size.toDouble / n.toDouble) -> n
  }.sorted.map(_.swap)

  result.takeRight(10) foreach { case (n, k) =>
    println(s"$n -> $k")
  }
}

Result:

36 -> 0.19444444444444445
10 -> 0.2
20 -> 0.2
30 -> 0.2
18 -> 0.2222222222222222
4 -> 0.25
8 -> 0.25
24 -> 0.25
6 -> 0.3333333333333333
12 -> 0.3333333333333333