160130

Ruby


引き算魔方陣

Excel VBA でパズルを解こう プログラムで脳トレに挑戦
という本に引き算魔方陣の問題が載っていた。

例えば、
2　1　4
3　5　7
6　9　8
がある。
これを[2, 1, 4, 3, 5, 7, 6, 9, 8] と表すこととし、
他の引き算魔方陣も出力してみた。

# -*- coding: cp932 -*-

def f(i, j, ary)
  ary[i] + ary[j] - ary[(i + j) / 2]
end

ary = []
(1..9).to_a.permutation(9){|p|
  i = f(0, 2, p)
  # 縦横斜めの各列で、両端の数字の合計から真ん中の数字を引いた値がどれも同じ
  if i == f(3, 5, p)
    if i == f(6, 8, p)
      if i == f(0, 6, p)
        if i == f(1, 7, p)
          if i == f(2, 8, p)
            if i == f(0, 8, p)
              if i == f(2, 6, p)
                ary << p
              end
            end
          end
        end
      end
    end
  end
}
p ary
puts "#{ary.size}通りある。"

出力結果
[[2, 1, 4, 3, 5, 7, 6, 9, 8], [2, 3, 6, 1, 5, 9, 4, 7, 8], [4, 1, 2, 7, 5, 3, 8, 9, 6], [4, 7, 8, 1, 5, 9, 2, 3, 6], [6, 3, 2, 9, 5, 1, 8, 7, 4], [6, 9, 8, 3, 5, 7, 2, 1, 4], [8, 7, 4, 9, 5, 1, 6, 3, 2], [8, 9, 6, 7, 5, 3, 4, 1, 2]]
8通りある。

160128

Ruby


誤っているようで誤っていない数式

Excel VBA でパズルを解こう プログラムで脳トレに挑戦
という本に次のような問題が載っていた。

(1) / (2) + (3) / (4) - (5) / (6)
= ((1) + (3) - (5)) / ((2) + (4) - (6))
（ただし、(1) ～ (6) は１～９までの異なる六つの数字）

例えば、
7 / 4 + 6 / 8 - 3 / 2 = (7 + 6 - 3) / (4 + 8 - 2)
が条件を満たす。

条件を満たす[(1), (2), (3), (4), (5), (6)] をすべて出力してみた。

# -*- coding: cp932 -*-

ary = []
(1..9).to_a.permutation(6){|p|
  if p[1] + p[3] - p[5] != 0
    ary << p if p[0].to_r / p[1] + p[2].to_r / p[3] - p[4].to_r / p[5] == (p[0] + p[2] - p[4]).to_r / (p[1] + p[3] - p[5])
  end
}
p ary
puts "#{ary.size}通りある。"

出力結果
[[1, 2, 3, 6, 5, 4], [1, 2, 3, 6, 7, 4], [1, 2, 3, 6, 8, 4], [1, 2, 3, 6, 9, 4], [1, 2, 4, 8, 3, 5], [1, 2, 4, 8, 3, 6], [1, 2, 4, 8, 6, 5], [1, 2, 4, 8, 7, 5], [1, 2, 4, 8, 9, 5], [1, 2, 5, 3, 7, 6], [1, 3, 4, 8, 5, 6], [1, 4, 2, 8, 3, 6], [1, 4, 2, 8, 5, 6], [1, 4, 2, 8, 7, 6], [1, 4, 2, 8, 9, 6], [1, 4, 3, 2, 6, 8], [1, 6, 3, 9, 4, 8], [2, 1, 4, 8, 9, 3], [2, 1, 6, 9, 8, 3], [2, 1, 8, 4, 6, 3], [2, 3, 4, 8, 1, 6], [2, 4, 3, 6, 1, 5], [2, 4, 3, 6, 7, 5], [2, 4, 3, 6, 8, 5], [2, 4, 3, 6, 9, 5], [2, 4, 3, 9, 5, 6], [2, 8, 1, 4, 3, 6], [2, 8, 1, 4, 5, 6], [2, 8, 1, 4, 7, 6], [2, 8, 1, 4, 9, 6], [3, 1, 4, 8, 7, 2], [3, 1, 6, 4, 9, 2], [3, 1, 9, 6, 7, 2], [3, 2, 1, 4, 6, 8], [3, 2, 6, 8, 9, 4], [3, 2, 6, 9, 4, 8], [3, 2, 9, 6, 1, 4], [3, 2, 9, 6, 5, 4], [3, 2, 9, 6, 7, 4], [3, 2, 9, 6, 8, 4], [3, 6, 1, 2, 5, 4], [3, 6, 1, 2, 7, 4], [3, 6, 1, 2, 8, 4], [3, 6, 1, 2, 9, 4], [3, 6, 2, 4, 1, 5], [3, 6, 2, 4, 7, 5], [3, 6, 2, 4, 8, 5], [3, 6, 2, 4, 9, 5], [3, 6, 4, 8, 1, 2], [3, 6, 4, 8, 1, 7], [3, 6, 4, 8, 2, 7], [3, 6, 4, 8, 5, 7], [3, 6, 4, 8, 9, 7], [3, 9, 1, 6, 4, 8], [3, 9, 2, 4, 5, 6], [4, 2, 8, 6, 7, 3], [4, 8, 1, 2, 3, 5], [4, 8, 1, 2, 3, 6], [4, 8, 1, 2, 6, 5], [4, 8, 1, 2, 7, 5], [4, 8, 1, 2, 9, 5], [4, 8, 1, 3, 5, 6], [4, 8, 2, 1, 9, 3], [4, 8, 2, 3, 1, 6], [4, 8, 3, 1, 7, 2], [4, 8, 3, 6, 1, 2], [4, 8, 3, 6, 1, 7], [4, 8, 3, 6, 2, 7], [4, 8, 3, 6, 5, 7], [4, 8, 3, 6, 9, 7], [4, 8, 5, 2, 9, 3], [4, 8, 5, 3, 6, 9], [5, 2, 4, 8, 9, 3], [5, 3, 1, 2, 7, 6], [5, 3, 4, 8, 6, 9], [6, 3, 8, 4, 2, 1], [6, 4, 3, 1, 9, 2], [6, 4, 7, 2, 3, 1], [6, 4, 7, 3, 2, 1], [6, 4, 9, 3, 5, 2], [6, 8, 3, 2, 9, 4], [6, 8, 7, 4, 3, 2], [6, 9, 2, 1, 8, 3], [6, 9, 3, 2, 4, 8], [6, 9, 7, 3, 2, 1], [6, 9, 7, 3, 2, 8], [7, 2, 6, 4, 3, 1], [7, 3, 6, 4, 2, 1], [7, 3, 6, 9, 2, 1], [7, 3, 6, 9, 2, 8], [7, 4, 6, 8, 3, 2], [8, 4, 2, 1, 6, 3], [8, 4, 6, 3, 2, 1], [8, 6, 4, 2, 7, 3], [9, 3, 6, 4, 5, 2], [9, 6, 3, 1, 7, 2], [9, 6, 3, 2, 1, 4], [9, 6, 3, 2, 5, 4], [9, 6, 3, 2, 7, 4], [9, 6, 3, 2, 8, 4]]
100通りある。

160126

Ruby


「和の２乗 = ３乗の和」となる組合せ(4)

1≦s1≦s2≦･･･≦s(n-1)≦sn≦n

なる自然数の組[s1,s2,･･･,s(n-1),sn]
において関係式
2(s1+s2+･･･+s(n-1)+sn)^4
=(s1^5+s2^5+･･･+s(n-1)^5+sn^5)+(s1^7+s2^7+･･･+s(n-1)^7+sn^7)
が成立する組合せ
についてもう少し調べてみた。

(2..18).each{|n|
  a_ary = []
  (1..n).to_a.repeated_combination(n){|c|
    a_ary << c if (c.inject(:+) ** 4) * 2 == c.map{|i| i ** 5}.inject(:+) + c.map{|i| i ** 7}.inject(:+)
  }
  p a_ary
}

出力結果
[[1, 2]]
[[1, 2, 3]]
[[1, 2, 3, 4]]
[[1, 2, 3, 4, 5]]
[[1, 2, 3, 4, 5, 6]]
[[1, 2, 3, 4, 5, 6, 7]]
[[1, 2, 3, 4, 5, 6, 7, 8]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]]
[[1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 7, 9, 9, 10, 12], [1, 1, 1, 2, 3, 4, 4, 5, 7, 7, 8, 8, 8, 11, 12, 14], [1, 1, 1, 7, 8, 9, 10, 11, 11, 14, 14, 15, 16, 16, 16, 16], [1, 2, 2, 3, 3, 4, 6, 8, 9, 12, 12, 12, 12, 12, 13, 16], [1, 2, 3, 4, 4, 5, 6, 9, 9, 9, 11, 12, 14, 14, 15, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [2, 2, 2, 2, 4, 4, 4, 5, 7, 7, 8, 10, 12, 13, 13, 14], [2, 4, 4, 4, 4, 4, 6, 6, 7, 7, 9, 9, 10, 10, 12, 16], [4, 4, 4, 4, 6, 6, 6, 6, 9, 9, 10, 10, 11, 12, 16, 16]]
[[1, 2, 2, 3, 4, 5, 6, 6, 8, 9, 10, 11, 11, 11, 11, 13, 17], [1, 2, 3, 3, 3, 5, 8, 8, 8, 9, 9, 9, 10, 11, 12, 16, 16], [1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 9, 9, 11, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 4, 4, 4, 4, 6, 6, 6, 13, 14, 14, 14, 15, 16, 16, 16, 16], [2, 2, 3, 3, 5, 5, 5, 5, 10, 11, 11, 14, 14, 14, 14, 16, 16], [2, 2, 3, 6, 6, 6, 8, 8, 10, 12, 13, 14, 16, 16, 16, 16, 17], [4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 7, 8, 10, 10, 10, 16], [4, 4, 4, 4, 5, 5, 5, 7, 8, 12, 12, 14, 15, 15, 15, 16, 17]]
[[1, 1, 1, 1, 1, 2, 3, 6, 6, 6, 7, 7, 10, 11, 11, 13, 13, 15], [1, 1, 1, 2, 2, 3, 4, 4, 5, 5, 7, 7, 8, 8, 8, 8, 10, 15], [1, 1, 1, 2, 3, 3, 4, 6, 7, 10, 10, 10, 11, 12, 12, 13, 16, 16], [1, 1, 1, 3, 3, 4, 4, 5, 7, 7, 11, 12, 12, 12, 12, 13, 14, 17], [1, 1, 1, 3, 5, 5, 7, 8, 10, 10, 10, 11, 15, 15, 15, 16, 16, 17], [1, 1, 1, 4, 6, 6, 6, 6, 7, 7, 7, 7, 9, 9, 11, 11, 14, 17], [1, 1, 2, 2, 2, 2, 2, 6, 6, 6, 10, 10, 10, 11, 13, 13, 15, 15], [1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 8, 10, 11, 12, 13], [1, 1, 2, 2, 2, 3, 3, 7, 7, 9, 10, 11, 12, 12, 12, 13, 14, 17], [1, 1, 2, 2, 4, 4, 4, 6, 7, 9, 9, 11, 11, 11, 12, 12, 15, 17], [1, 1, 2, 3, 4, 8, 8, 8, 8, 8, 8, 14, 14, 15, 15, 16, 16, 17], [1, 1, 2, 4, 4, 4, 7, 7, 7, 8, 9, 12, 12, 13, 15, 16, 16, 16], [1, 1, 2, 4, 4, 4, 7, 7, 7, 8, 10, 10, 11, 12, 13, 13, 13, 18], [1, 1, 3, 3, 5, 7, 8, 12, 12, 14, 14, 15, 15, 16, 17, 17, 18, 18], [1, 1, 3, 4, 4, 4, 4, 5, 6, 6, 7, 7, 8, 10, 11, 13, 14, 16], [1, 1, 3, 4, 5, 7, 9, 10, 11, 11, 14, 14, 14, 16, 16, 17, 18, 18], [1, 2, 2, 2, 4, 4, 4, 4, 6, 6, 6, 7, 10, 10, 10, 10, 13, 16], [1, 2, 2, 3, 4, 4, 8, 8, 11, 11, 11, 11, 13, 14, 14, 17, 17, 17], [1, 2, 3, 3, 4, 5, 7, 7, 8, 8, 9, 9, 9, 9, 10, 15, 15, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 3, 3, 3, 4, 7, 7, 8, 9, 9, 14, 14, 14, 15, 16, 17, 17, 17], [1, 3, 3, 3, 6, 8, 8, 8, 8, 8, 8, 9, 12, 12, 12, 13, 17, 18], [1, 3, 4, 4, 4, 6, 10, 14, 14, 14, 14, 15, 16, 17, 18, 18, 18, 18], [1, 4, 4, 4, 4, 4, 5, 6, 6, 10, 10, 14, 14, 15, 16, 16, 16, 16], [2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 5, 6, 6, 10, 10, 11, 14, 14], [2, 2, 2, 2, 3, 4, 5, 6, 6, 10, 10, 10, 10, 11, 13, 14, 15, 17], [2, 2, 2, 2, 5, 6, 6, 6, 7, 8, 8, 10, 10, 12, 13, 16, 16, 16], [2, 2, 2, 5, 6, 7, 7, 7, 8, 9, 10, 11, 12, 12, 15, 17, 17, 17], [2, 2, 3, 4, 4, 6, 6, 6, 6, 8, 9, 13, 14, 14, 14, 14, 14, 18], [2, 2, 4, 4, 4, 4, 5, 6, 7, 9, 12, 14, 14, 14, 15, 16, 16, 17], [2, 3, 3, 4, 6, 8, 9, 9, 9, 9, 9, 14, 15, 15, 16, 16, 18, 18], [2, 3, 4, 4, 5, 5, 7, 7, 8, 8, 8, 9, 9, 10, 10, 12, 15, 18], [2, 3, 4, 6, 6, 7, 7, 8, 12, 12, 12, 14, 16, 17, 17, 17, 18, 18], [2, 3, 5, 5, 5, 5, 6, 6, 6, 6, 7, 11, 12, 14, 14, 14, 14, 18], [2, 4, 4, 4, 4, 4, 6, 7, 10, 10, 12, 13, 13, 16, 16, 16, 16, 18], [2, 6, 6, 6, 6, 6, 6, 7, 7, 9, 9, 12, 15, 15, 16, 17, 17, 18], [3, 3, 4, 4, 6, 6, 7, 7, 12, 12, 12, 13, 15, 17, 17, 17, 17, 18], [3, 3, 4, 6, 7, 9, 9, 9, 9, 9, 9, 11, 11, 13, 15, 18, 18, 18], [3, 4, 4, 4, 5, 5, 6, 7, 8, 11, 13, 14, 14, 14, 14, 16, 18, 18], [3, 5, 5, 5, 7, 7, 7, 13, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18]]

160124

Ruby


「和の２乗 = ３乗の和」となる組合せ(3)

http://6626.teacup.com/shochandas/bbs/12291
において、
1≦s2≦･･･≦s(n-1)≦n
なる自然数の組[1,s2,･･･,s(n-1),n]
において関係式
(1+s2+･･･+s(n-1)+n)^2=1^3+s2^3+･･･+s(n-1)^3+n^3

かつ
2(1+s2+･･･+s(n-1)+n)^4
=(1^5+s2^5+･･･+s(n-1)^5+n^5)+(1^7+s2^7+･･･+s(n-1)^7+n^7)
かつ
16(1+s2+･･･+s(n-1)+n)^5
=(1^5+s2^5+･･･+s(n-1)^5+n^5)+10(1^7+s2^7+･･･+s(n-1)^7+n^7)+5(1^9+s2^9+･･･+s(n-1)^9+n^9)
が成立する組合せ
も探して欲しいとあったのですが、

条件を緩めて次を探すことにした。

1≦s1≦s2≦･･･≦s(n-1)≦sn≦n
なる自然数の組[s1,s2,･･･,s(n-1),sn]
において関係式
2(s1+s2+･･･+s(n-1)+sn)^4
=(s1^5+s2^5+･･･+s(n-1)^5+sn^5)+(s1^7+s2^7+･･･+s(n-1)^7+sn^7)
または
16(s1+s2+･･･+s(n-1)+sn)^5
=(s1^5+s2^5+･･･+s(n-1)^5+sn^5)+10(s1^7+s2^7+･･･+s(n-1)^7+sn^7)+5(s1^9+s2^9+･･･+s(n-1)^9+sn^9)
が成立する組合せ

(2..16).each{|n|
  a_ary = []
  (1..n).to_a.repeated_combination(n){|c|
    a_ary << c if (c.inject(:+) ** 4) * 2 == c.map{|i| i ** 5}.inject(:+) + c.map{|i| i ** 7}.inject(:+)
  }
  p a_ary
}

(2..16).each{|n|
  a_ary = []
  (1..n).to_a.repeated_combination(n){|c|
    a_ary << c if (c.inject(:+) ** 5) * 16 == c.map{|i| i ** 5}.inject(:+) + (c.map{|i| i ** 7}.inject(:+)) * 10 + (c.map{|i| i ** 9}.inject(:+)) * 5
  }
  p a_ary
}

出力結果
[[1, 2]]
[[1, 2, 3]]
[[1, 2, 3, 4]]
[[1, 2, 3, 4, 5]]
[[1, 2, 3, 4, 5, 6]]
[[1, 2, 3, 4, 5, 6, 7]]
[[1, 2, 3, 4, 5, 6, 7, 8]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]]
[[1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 7, 9, 9, 10, 12], [1, 1, 1, 2, 3, 4, 4, 5, 7, 7, 8, 8, 8, 11, 12, 14], [1, 1, 1, 7, 8, 9, 10, 11, 11, 14, 14, 15, 16, 16, 16, 16], [1, 2, 2, 3, 3, 4, 6, 8, 9, 12, 12, 12, 12, 12, 13, 16], [1, 2, 3, 4, 4, 5, 6, 9, 9, 9, 11, 12, 14, 14, 15, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [2, 2, 2, 2, 4, 4, 4, 5, 7, 7, 8, 10, 12, 13, 13, 14], [2, 4, 4, 4, 4, 4, 6, 6, 7, 7, 9, 9, 10, 10, 12, 16], [4, 4, 4, 4, 6, 6, 6, 6, 9, 9, 10, 10, 11, 12, 16, 16]]
[[1, 2]]
[[1, 2, 3]]
[[1, 2, 3, 4]]
[[1, 2, 3, 4, 5]]
[[1, 2, 3, 4, 5, 6]]
[[1, 2, 3, 4, 5, 6, 7]]
[[1, 2, 3, 4, 5, 6, 7, 8]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]]
[[1, 1, 2, 2, 3, 3, 3, 3, 6, 6, 9, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]]
[[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]]

160123(2)

「和の２乗 = ３乗の和」となる組合せ(2)

以下のような例を見つけた。

（証明は簡単なので省略しました。）

n が3 以上のとき
1～n からn - 1 を抜き2 を入れる。
例
1 2 3
1 2 2 4
1 2 2 3 5
1 2 2 3 4 6
1 2 2 3 4 5 7

n が4 以上の偶数のとき
1 が1 個
n - 2 がn / 2 個
n がn / 2 - 1 個
例
1 2 2 4
1 4 4 4 6 6
1 6 6 6 6 8 8 8

160123

Ruby


「和の２乗 = ３乗の和」となる組合せ(1)

http://6626.teacup.com/shochandas/bbs/12272
において、
1≦s2≦s3≦s4≦･･･≦s(n-1)≦n
なる自然数の組[1,s2,s3,s4,･･･,s(n-1),n]
において関係式
(1+s2+s3+s4+･･･+s(n-1)+n)^2=1^3+s2^3+s3^3+s4^3+･･･+s(n-1)^3+n^3
が成立する組合せ
を発見して欲しいとあり、n が2以上12以下の場合についてすべて求めてみた。

(2..12).each{|n|
  a_ary = []
  (1..n).to_a.repeated_combination(n){|c|
    # 最初が1、最後がn
    if c[0] == 1 && c[-1] == n
      a_ary << c if c.inject(:+) ** 2 == c.map{|i| i * i * i}.inject(:+)
    end
  }
  p a_ary
}

出力結果
[[1, 2]]
[[1, 2, 3]]
[[1, 2, 2, 4], [1, 2, 3, 4]]
[[1, 2, 2, 3, 5], [1, 2, 3, 4, 5]]
[[1, 2, 2, 3, 4, 6], [1, 2, 2, 4, 4, 6], [1, 2, 3, 4, 5, 6], [1, 4, 4, 4, 6, 6]]
[[1, 1, 3, 4, 4, 5, 7], [1, 1, 4, 5, 5, 5, 7], [1, 2, 2, 3, 3, 3, 7], [1, 2, 2, 3, 4, 5, 7], [1, 2, 3, 4, 5, 6, 7], [1, 3, 5, 5, 5, 7, 7], [1, 4, 4, 5, 6, 7, 7], [1, 5, 5, 6, 6, 7, 7]]
[[1, 1, 2, 4, 4, 5, 5, 8], [1, 1, 2, 4, 5, 5, 5, 8], [1, 1, 3, 3, 3, 4, 5, 8], [1, 1, 4, 5, 5, 5, 7, 8], [1, 2, 2, 2, 4, 4, 4, 8], [1, 2, 2, 3, 4, 4, 6, 8], [1, 2, 2, 3, 4, 5, 6, 8], [1, 2, 2, 4, 4, 6, 6, 8], [1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 4, 5, 5, 7, 7, 8], [1, 2, 5, 6, 6, 7, 7, 8], [1, 3, 3, 3, 3, 5, 7, 8], [1, 3, 3, 3, 6, 6, 7, 8], [1, 3, 4, 4, 4, 5, 8, 8], [1, 3, 4, 4, 5, 6, 8, 8], [1, 3, 4, 6, 6, 6, 8, 8], [1, 3, 4, 6, 7, 7, 7, 8], [1, 4, 5, 5, 7, 7, 8, 8], [1, 4, 5, 6, 7, 7, 8, 8], [1, 5, 5, 7, 7, 7, 8, 8], [1, 6, 6, 6, 6, 8, 8, 8]]
[[1, 1, 1, 4, 5, 5, 5, 5, 9], [1, 1, 2, 3, 4, 5, 5, 6, 9], [1, 1, 2, 4, 5, 5, 5, 7, 9], [1, 1, 2, 5, 5, 6, 6, 7, 9], [1, 1, 3, 3, 3, 3, 5, 5, 9], [1, 1, 3, 3, 3, 4, 4, 6, 9], [1, 1, 3, 3, 3, 5, 6, 6, 9], [1, 1, 3, 4, 5, 6, 7, 7, 9], [1, 1, 4, 5, 5, 5, 7, 8, 9], [1, 1, 4, 5, 5, 6, 7, 8, 9], [1, 1, 5, 5, 5, 7, 7, 8, 9], [1, 1, 6, 6, 7, 7, 7, 8, 9], [1, 2, 2, 2, 3, 5, 5, 5, 9], [1, 2, 2, 3, 3, 4, 6, 6, 9], [1, 2, 2, 3, 4, 4, 5, 7, 9], [1, 2, 2, 3, 4, 5, 6, 7, 9], [1, 2, 2, 3, 6, 6, 6, 7, 9], [1, 2, 2, 5, 5, 5, 6, 8, 9], [1, 2, 2, 5, 5, 7, 7, 7, 9], [1, 2, 2, 5, 6, 7, 7, 7, 9], [1, 2, 3, 3, 3, 3, 3, 3, 9], [1, 2, 3, 3, 3, 3, 3, 6, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 6, 6, 7, 8, 9], [1, 2, 3, 5, 6, 7, 7, 8, 9], [1, 2, 4, 4, 4, 4, 7, 8, 9], [1, 2, 4, 5, 6, 6, 6, 9, 9], [1, 2, 4, 6, 6, 7, 8, 8, 9], [1, 2, 5, 5, 5, 6, 7, 9, 9], [1, 2, 5, 7, 7, 7, 8, 8, 9], [1, 3, 3, 3, 3, 5, 5, 8, 9], [1, 3, 3, 4, 6, 6, 8, 8, 9], [1, 3, 3, 5, 5, 7, 8, 8, 9], [1, 3, 4, 5, 7, 7, 7, 9, 9], [1, 3, 4, 6, 7, 8, 8, 8, 9], [1, 3, 5, 6, 7, 7, 8, 9, 9], [1, 3, 6, 7, 8, 8, 8, 8, 9], [1, 4, 4, 4, 4, 7, 7, 9, 9], [1, 4, 4, 7, 8, 8, 8, 8, 9], [1, 4, 5, 6, 6, 6, 9, 9, 9], [1, 5, 5, 5, 7, 7, 9, 9, 9], [1, 5, 6, 6, 7, 8, 9, 9, 9]]
[[1, 1, 1, 4, 4, 4, 4, 5, 6, 10], [1, 1, 1, 4, 4, 5, 5, 5, 7, 10], [1, 1, 1, 4, 6, 6, 6, 7, 7, 10], [1, 1, 2, 2, 4, 4, 5, 5, 5, 10], [1, 1, 2, 2, 4, 5, 5, 6, 6, 10], [1, 1, 2, 3, 4, 4, 5, 6, 7, 10], [1, 1, 2, 4, 4, 5, 5, 5, 8, 10], [1, 1, 2, 4, 4, 5, 5, 6, 8, 10], [1, 1, 2, 4, 5, 5, 5, 7, 8, 10], [1, 1, 2, 4, 6, 6, 7, 7, 8, 10], [1, 1, 2, 6, 6, 6, 7, 8, 8, 10], [1, 1, 3, 5, 5, 5, 5, 7, 9, 10], [1, 1, 3, 5, 6, 7, 8, 8, 8, 10], [1, 1, 4, 4, 5, 5, 8, 8, 8, 10], [1, 1, 4, 5, 5, 5, 7, 8, 9, 10], [1, 1, 4, 5, 5, 8, 8, 8, 8, 10], [1, 1, 4, 6, 7, 7, 8, 8, 9, 10], [1, 1, 4, 7, 8, 8, 8, 8, 8, 10], [1, 1, 5, 5, 5, 6, 6, 6, 10, 10], [1, 1, 6, 6, 6, 6, 7, 8, 10, 10], [1, 2, 2, 2, 3, 4, 4, 4, 5, 10], [1, 2, 2, 2, 3, 4, 4, 5, 6, 10], [1, 2, 2, 2, 3, 4, 6, 6, 6, 10], [1, 2, 2, 2, 4, 5, 5, 7, 7, 10], [1, 2, 2, 2, 4, 5, 6, 7, 7, 10], [1, 2, 2, 2, 5, 5, 7, 7, 7, 10], [1, 2, 2, 2, 6, 6, 6, 6, 8, 10], [1, 2, 2, 3, 4, 4, 5, 6, 8, 10], [1, 2, 2, 3, 4, 4, 6, 6, 8, 10], [1, 2, 2, 3, 4, 5, 6, 7, 8, 10], [1, 2, 2, 4, 4, 4, 4, 4, 8, 10], [1, 2, 2, 4, 4, 6, 6, 8, 8, 10], [1, 2, 2, 4, 5, 5, 5, 6, 9, 10], [1, 2, 2, 4, 5, 7, 7, 8, 8, 10], [1, 2, 2, 5, 5, 7, 7, 7, 9, 10], [1, 2, 2, 5, 7, 7, 8, 8, 8, 10], [1, 2, 2, 6, 6, 6, 7, 8, 9, 10], [1, 2, 3, 3, 3, 3, 3, 3, 5, 10], [1, 2, 3, 3, 4, 5, 5, 8, 8, 10], [1, 2, 3, 3, 5, 5, 6, 7, 9, 10], [1, 2, 3, 4, 4, 5, 7, 7, 9, 10], [1, 2, 3, 4, 4, 6, 8, 8, 8, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 6, 8, 8, 8, 8, 10], [1, 2, 3, 5, 6, 6, 6, 6, 10, 10], [1, 2, 3, 5, 7, 7, 8, 8, 9, 10], [1, 2, 3, 5, 8, 8, 8, 8, 8, 10], [1, 2, 4, 4, 4, 4, 4, 7, 9, 10], [1, 2, 4, 4, 4, 8, 8, 8, 8, 10], [1, 2, 4, 5, 5, 5, 5, 7, 10, 10], [1, 2, 4, 5, 5, 5, 7, 9, 9, 10], [1, 2, 4, 5, 6, 6, 8, 9, 9, 10], [1, 2, 5, 5, 5, 5, 7, 8, 10, 10], [1, 2, 5, 5, 7, 7, 8, 8, 10, 10], [1, 2, 5, 6, 6, 6, 9, 9, 9, 10], [1, 2, 5, 6, 6, 7, 9, 9, 9, 10], [1, 2, 5, 7, 7, 8, 8, 8, 10, 10], [1, 2, 6, 6, 6, 8, 9, 9, 9, 10], [1, 2, 6, 6, 8, 8, 8, 8, 10, 10], [1, 3, 3, 3, 3, 3, 3, 7, 7, 10], [1, 3, 3, 3, 3, 3, 5, 7, 8, 10], [1, 3, 3, 3, 3, 5, 5, 6, 9, 10], [1, 3, 3, 3, 3, 5, 7, 8, 8, 10], [1, 3, 3, 3, 4, 7, 8, 8, 8, 10], [1, 3, 3, 3, 5, 5, 7, 8, 9, 10], [1, 3, 3, 4, 4, 4, 6, 8, 9, 10], [1, 3, 3, 4, 6, 7, 7, 7, 10, 10], [1, 3, 3, 4, 6, 8, 8, 8, 9, 10], [1, 3, 3, 5, 5, 6, 6, 8, 10, 10], [1, 3, 3, 5, 7, 7, 7, 8, 10, 10], [1, 3, 3, 6, 6, 8, 8, 9, 9, 10], [1, 3, 3, 6, 7, 8, 8, 9, 9, 10], [1, 3, 4, 4, 4, 7, 7, 7, 10, 10], [1, 3, 4, 4, 4, 7, 7, 9, 9, 10], [1, 3, 4, 5, 6, 6, 7, 9, 10, 10], [1, 3, 4, 5, 6, 8, 8, 8, 10, 10], [1, 3, 4, 5, 7, 8, 8, 8, 10, 10], [1, 3, 4, 6, 8, 8, 8, 8, 10, 10], [1, 3, 5, 5, 5, 5, 7, 9, 10, 10], [1, 3, 5, 5, 8, 8, 9, 9, 9, 10], [1, 3, 5, 7, 8, 8, 8, 9, 10, 10], [1, 3, 6, 6, 7, 7, 7, 10, 10, 10], [1, 3, 6, 7, 7, 8, 9, 9, 10, 10], [1, 3, 6, 8, 8, 9, 9, 9, 9, 10], [1, 4, 4, 4, 4, 4, 4, 6, 10, 10], [1, 4, 4, 4, 5, 6, 9, 9, 9, 10], [1, 4, 4, 6, 7, 7, 9, 9, 10, 10], [1, 4, 4, 7, 8, 9, 9, 9, 9, 10], [1, 4, 5, 5, 5, 9, 9, 9, 9, 10], [1, 4, 5, 7, 7, 8, 8, 10, 10, 10], [1, 4, 5, 8, 8, 8, 9, 9, 10, 10], [1, 4, 6, 6, 6, 9, 9, 9, 10, 10], [1, 4, 6, 8, 8, 8, 8, 10, 10, 10], [1, 4, 8, 8, 8, 8, 8, 10, 10, 10], [1, 5, 5, 5, 5, 5, 7, 10, 10, 10], [1, 5, 5, 5, 5, 7, 8, 10, 10, 10], [1, 5, 5, 9, 9, 9, 9, 9, 9, 10], [1, 8, 8, 8, 8, 8, 10, 10, 10, 10], [1, 8, 8, 8, 9, 9, 9, 10, 10, 10]]
[[1, 1, 1, 2, 4, 5, 5, 6, 6, 6, 11], [1, 1, 1, 2, 5, 5, 5, 5, 6, 7, 11], [1, 1, 1, 3, 4, 5, 5, 7, 7, 7, 11], [1, 1, 1, 3, 5, 5, 6, 7, 7, 8, 11], [1, 1, 1, 3, 6, 6, 6, 6, 8, 8, 11], [1, 1, 1, 4, 4, 4, 4, 5, 7, 7, 11], [1, 1, 1, 4, 5, 5, 6, 6, 6, 9, 11], [1, 1, 1, 5, 5, 5, 5, 7, 7, 9, 11], [1, 1, 1, 5, 5, 6, 6, 7, 8, 9, 11], [1, 1, 2, 2, 3, 4, 6, 6, 6, 6, 11], [1, 1, 2, 2, 3, 5, 5, 6, 6, 7, 11], [1, 1, 2, 2, 4, 6, 6, 6, 7, 8, 11], [1, 1, 2, 2, 5, 5, 5, 5, 7, 8, 11], [1, 1, 2, 2, 6, 6, 6, 7, 8, 8, 11], [1, 1, 2, 3, 3, 4, 4, 5, 5, 7, 11], [1, 1, 2, 3, 3, 4, 6, 7, 7, 7, 11], [1, 1, 2, 3, 3, 6, 6, 7, 7, 8, 11], [1, 1, 2, 3, 4, 4, 5, 5, 7, 8, 11], [1, 1, 2, 3, 4, 6, 6, 6, 6, 9, 11], [1, 1, 2, 3, 5, 5, 6, 7, 7, 9, 11], [1, 1, 2, 3, 6, 6, 6, 7, 8, 9, 11], [1, 1, 2, 4, 4, 4, 5, 7, 8, 8, 11], [1, 1, 2, 4, 4, 5, 5, 5, 7, 9, 11], [1, 1, 2, 4, 5, 5, 5, 7, 8, 9, 11], [1, 1, 2, 4, 5, 7, 7, 8, 8, 9, 11], [1, 1, 2, 4, 7, 7, 8, 8, 8, 9, 11], [1, 1, 2, 5, 5, 5, 6, 6, 6, 10, 11], [1, 1, 2, 5, 6, 6, 6, 7, 8, 10, 11], [1, 1, 2, 5, 6, 6, 7, 7, 8, 10, 11], [1, 1, 2, 6, 6, 6, 7, 8, 8, 10, 11], [1, 1, 2, 7, 7, 7, 7, 9, 9, 9, 11], [1, 1, 3, 3, 3, 3, 3, 5, 6, 6, 11], [1, 1, 3, 3, 3, 3, 3, 6, 6, 6, 11], [1, 1, 3, 3, 3, 3, 4, 4, 4, 6, 11], [1, 1, 3, 3, 3, 3, 6, 6, 6, 8, 11], [1, 1, 3, 3, 3, 4, 4, 5, 5, 8, 11], [1, 1, 3, 3, 3, 5, 6, 6, 6, 9, 11], [1, 1, 3, 3, 3, 5, 7, 7, 8, 8, 11], [1, 1, 3, 3, 4, 5, 6, 6, 8, 9, 11], [1, 1, 3, 3, 4, 7, 7, 7, 8, 9, 11], [1, 1, 3, 3, 6, 6, 6, 7, 7, 10, 11], [1, 1, 3, 4, 4, 4, 4, 4, 8, 8, 11], [1, 1, 3, 4, 4, 4, 8, 8, 8, 8, 11], [1, 1, 3, 4, 4, 6, 6, 7, 9, 9, 11], [1, 1, 3, 4, 4, 6, 8, 8, 8, 9, 11], [1, 1, 3, 4, 4, 7, 8, 8, 8, 9, 11], [1, 1, 3, 4, 5, 5, 5, 7, 9, 9, 11], [1, 1, 3, 4, 6, 6, 6, 8, 8, 10, 11], [1, 1, 3, 4, 7, 8, 8, 8, 9, 9, 11], [1, 1, 3, 5, 5, 6, 7, 9, 9, 9, 11], [1, 1, 3, 5, 5, 7, 8, 8, 8, 10, 11], [1, 1, 3, 6, 6, 6, 8, 8, 9, 10, 11], [1, 1, 3, 6, 7, 8, 8, 8, 9, 10, 11], [1, 1, 4, 4, 4, 4, 6, 6, 7, 10, 11], [1, 1, 4, 4, 4, 5, 5, 5, 5, 10, 11], [1, 1, 4, 4, 4, 5, 6, 7, 8, 10, 11], [1, 1, 4, 4, 7, 7, 8, 8, 9, 10, 11], [1, 1, 4, 4, 8, 8, 8, 9, 9, 9, 11], [1, 1, 4, 5, 5, 5, 5, 7, 9, 10, 11], [1, 1, 4, 5, 5, 5, 7, 8, 9, 10, 11], [1, 1, 4, 5, 5, 6, 6, 6, 7, 11, 11], [1, 1, 4, 5, 5, 6, 6, 7, 7, 11, 11], [1, 1, 4, 5, 6, 6, 6, 7, 8, 11, 11], [1, 1, 4, 6, 6, 6, 7, 8, 10, 10, 11], [1, 1, 5, 5, 5, 5, 5, 6, 6, 11, 11], [1, 1, 5, 5, 5, 6, 6, 6, 10, 10, 11], [1, 1, 5, 5, 6, 6, 7, 7, 9, 11, 11], [1, 1, 5, 5, 6, 7, 9, 9, 9, 10, 11], [1, 1, 5, 5, 7, 7, 7, 8, 9, 11, 11], [1, 1, 5, 5, 7, 8, 9, 9, 9, 10, 11], [1, 1, 5, 9, 9, 9, 9, 9, 9, 9, 11], [1, 1, 6, 6, 6, 6, 7, 9, 9, 11, 11], [1, 1, 6, 6, 6, 7, 8, 9, 9, 11, 11], [1, 1, 7, 7, 8, 8, 9, 9, 9, 11, 11], [1, 1, 7, 8, 8, 8, 8, 10, 10, 10, 11], [1, 1, 8, 8, 8, 9, 9, 9, 10, 10, 11], [1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 11], [1, 2, 2, 2, 3, 3, 6, 6, 6, 7, 11], [1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 11], [1, 2, 2, 2, 3, 4, 5, 7, 7, 7, 11], [1, 2, 2, 2, 3, 5, 5, 6, 7, 8, 11], [1, 2, 2, 2, 4, 4, 4, 5, 6, 8, 11], [1, 2, 2, 2, 4, 4, 7, 7, 7, 8, 11], [1, 2, 2, 2, 5, 5, 7, 7, 7, 9, 11], [1, 2, 2, 2, 7, 7, 7, 7, 8, 9, 11], [1, 2, 2, 3, 3, 3, 4, 4, 5, 7, 11], [1, 2, 2, 3, 3, 3, 4, 4, 6, 7, 11], [1, 2, 2, 3, 3, 3, 4, 5, 7, 7, 11], [1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 11], [1, 2, 2, 3, 3, 6, 6, 7, 7, 9, 11], [1, 2, 2, 3, 4, 4, 5, 6, 7, 9, 11], [1, 2, 2, 3, 4, 5, 6, 7, 8, 9, 11], [1, 2, 2, 3, 5, 5, 7, 8, 8, 9, 11], [1, 2, 2, 3, 5, 6, 6, 6, 6, 10, 11], [1, 2, 2, 4, 4, 4, 5, 7, 8, 9, 11], [1, 2, 2, 4, 4, 5, 6, 6, 6, 10, 11], [1, 2, 2, 4, 4, 6, 7, 7, 7, 10, 11], [1, 2, 2, 5, 5, 7, 7, 7, 9, 10, 11], [1, 2, 2, 5, 6, 6, 7, 8, 9, 10, 11], [1, 2, 2, 5, 6, 8, 8, 9, 9, 9, 11], [1, 2, 2, 6, 6, 6, 7, 7, 7, 11, 11], [1, 2, 3, 3, 3, 3, 4, 4, 6, 8, 11], [1, 2, 3, 3, 3, 4, 4, 7, 8, 8, 11], [1, 2, 3, 3, 3, 4, 5, 7, 7, 9, 11], [1, 2, 3, 3, 3, 4, 7, 8, 8, 8, 11], [1, 2, 3, 3, 4, 5, 6, 7, 9, 9, 11], [1, 2, 3, 3, 6, 6, 7, 9, 9, 9, 11], [1, 2, 3, 3, 6, 7, 8, 8, 8, 10, 11], [1, 2, 3, 3, 8, 8, 8, 8, 9, 9, 11], [1, 2, 3, 4, 4, 4, 5, 5, 6, 10, 11], [1, 2, 3, 4, 4, 5, 5, 7, 8, 10, 11], [1, 2, 3, 4, 4, 8, 8, 8, 9, 9, 11], [1, 2, 3, 4, 5, 5, 6, 7, 9, 10, 11], [1, 2, 3, 4, 5, 5, 7, 7, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 6, 6, 6, 6, 7, 11, 11], [1, 2, 3, 4, 6, 6, 6, 7, 7, 11, 11], [1, 2, 3, 5, 5, 5, 5, 5, 9, 10, 11], [1, 2, 3, 5, 5, 5, 6, 6, 6, 11, 11], [1, 2, 3, 5, 5, 7, 7, 7, 8, 11, 11], [1, 2, 3, 5, 5, 7, 7, 9, 9, 10, 11], [1, 2, 3, 5, 6, 6, 6, 6, 10, 10, 11], [1, 2, 3, 6, 6, 6, 7, 7, 9, 11, 11], [1, 2, 3, 6, 6, 7, 9, 9, 9, 10, 11], [1, 2, 3, 6, 7, 7, 7, 8, 9, 11, 11], [1, 2, 3, 6, 7, 8, 9, 9, 9, 10, 11], [1, 2, 3, 7, 9, 9, 9, 9, 9, 9, 11], [1, 2, 4, 4, 4, 5, 5, 9, 9, 9, 11], [1, 2, 4, 4, 4, 5, 8, 9, 9, 9, 11], [1, 2, 4, 4, 6, 9, 9, 9, 9, 9, 11], [1, 2, 4, 4, 7, 7, 7, 7, 9, 11, 11], [1, 2, 4, 4, 8, 9, 9, 9, 9, 9, 11], [1, 2, 4, 5, 5, 6, 6, 7, 9, 11, 11], [1, 2, 4, 5, 5, 7, 9, 9, 9, 10, 11], [1, 2, 4, 5, 6, 7, 8, 9, 10, 10, 11], [1, 2, 4, 5, 7, 8, 8, 9, 10, 10, 11], [1, 2, 4, 6, 6, 6, 7, 9, 9, 11, 11], [1, 2, 4, 6, 7, 8, 8, 9, 9, 11, 11], [1, 2, 4, 7, 7, 7, 7, 10, 10, 10, 11], [1, 2, 4, 8, 8, 8, 8, 9, 9, 11, 11], [1, 2, 4, 8, 9, 9, 9, 9, 9, 10, 11], [1, 2, 5, 5, 6, 8, 9, 9, 10, 10, 11], [1, 2, 5, 6, 6, 6, 6, 10, 10, 10, 11], [1, 2, 5, 6, 6, 6, 8, 10, 10, 10, 11], [1, 2, 5, 6, 6, 9, 9, 9, 10, 10, 11], [1, 2, 5, 7, 8, 8, 9, 10, 10, 10, 11], [1, 2, 5, 7, 9, 9, 9, 9, 10, 10, 11], [1, 2, 6, 6, 6, 6, 8, 9, 10, 11, 11], [1, 2, 7, 7, 7, 8, 8, 10, 10, 11, 11], [1, 2, 7, 7, 8, 9, 9, 9, 10, 11, 11], [1, 2, 7, 8, 8, 9, 9, 9, 10, 11, 11], [1, 3, 3, 3, 3, 3, 3, 3, 3, 6, 11], [1, 3, 3, 3, 3, 3, 4, 5, 6, 9, 11], [1, 3, 3, 3, 3, 4, 4, 4, 7, 9, 11], [1, 3, 3, 3, 3, 4, 4, 6, 8, 9, 11], [1, 3, 3, 3, 3, 4, 7, 8, 8, 9, 11], [1, 3, 3, 3, 3, 5, 5, 6, 7, 10, 11], [1, 3, 3, 3, 3, 5, 5, 6, 9, 9, 11], [1, 3, 3, 3, 3, 6, 6, 6, 8, 10, 11], [1, 3, 3, 3, 3, 6, 7, 8, 9, 9, 11], [1, 3, 3, 3, 3, 7, 7, 8, 9, 9, 11], [1, 3, 3, 3, 4, 4, 5, 5, 5, 10, 11], [1, 3, 3, 3, 4, 4, 5, 5, 9, 9, 11], [1, 3, 3, 3, 4, 5, 5, 5, 8, 10, 11], [1, 3, 3, 3, 4, 5, 7, 8, 8, 10, 11], [1, 3, 3, 3, 4, 6, 6, 9, 9, 9, 11], [1, 3, 3, 3, 4, 7, 8, 8, 8, 10, 11], [1, 3, 3, 3, 6, 6, 9, 9, 9, 9, 11], [1, 3, 3, 3, 7, 7, 7, 9, 9, 10, 11], [1, 3, 3, 3, 7, 8, 9, 9, 9, 9, 11], [1, 3, 3, 4, 4, 5, 6, 8, 9, 10, 11], [1, 3, 3, 4, 5, 6, 7, 8, 8, 11, 11], [1, 3, 3, 4, 5, 6, 8, 9, 9, 10, 11], [1, 3, 3, 4, 7, 8, 8, 8, 10, 10, 11], [1, 3, 3, 5, 5, 5, 6, 8, 10, 10, 11], [1, 3, 3, 5, 6, 6, 8, 9, 10, 10, 11], [1, 3, 3, 5, 7, 7, 7, 9, 9, 11, 11], [1, 3, 3, 5, 8, 8, 8, 9, 10, 10, 11], [1, 3, 3, 6, 8, 8, 9, 9, 10, 10, 11], [1, 3, 4, 4, 4, 4, 6, 6, 7, 11, 11], [1, 3, 4, 4, 4, 5, 5, 5, 7, 11, 11], [1, 3, 4, 4, 4, 7, 8, 8, 8, 11, 11], [1, 3, 4, 4, 5, 5, 6, 7, 9, 11, 11], [1, 3, 4, 4, 5, 5, 8, 8, 10, 10, 11], [1, 3, 4, 4, 5, 6, 7, 9, 10, 10, 11], [1, 3, 4, 5, 6, 7, 9, 9, 9, 11, 11], [1, 3, 4, 5, 8, 8, 8, 10, 10, 10, 11], [1, 3, 4, 7, 7, 7, 9, 9, 10, 11, 11], [1, 3, 5, 5, 7, 7, 9, 9, 10, 11, 11], [1, 3, 5, 6, 6, 7, 7, 8, 11, 11, 11], [1, 3, 5, 6, 7, 8, 8, 8, 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5, 5, 6, 9, 10, 12], [1, 3, 3, 3, 3, 4, 6, 6, 6, 6, 11, 12], [1, 3, 3, 3, 3, 4, 8, 9, 9, 9, 9, 12], [1, 3, 3, 3, 3, 5, 5, 6, 7, 8, 11, 12], [1, 3, 3, 3, 3, 5, 5, 6, 9, 9, 10, 12], [1, 3, 3, 3, 3, 5, 7, 8, 8, 10, 10, 12], [1, 3, 3, 3, 3, 6, 6, 6, 6, 9, 11, 12], [1, 3, 3, 3, 3, 6, 6, 7, 8, 9, 11, 12], [1, 3, 3, 3, 3, 7, 8, 8, 8, 9, 11, 12], [1, 3, 3, 3, 4, 4, 5, 7, 8, 8, 11, 12], [1, 3, 3, 3, 4, 4, 6, 7, 7, 9, 11, 12], [1, 3, 3, 3, 4, 5, 5, 5, 8, 10, 10, 12], [1, 3, 3, 3, 4, 6, 7, 7, 10, 10, 10, 12], [1, 3, 3, 3, 4, 7, 8, 8, 8, 10, 11, 12], [1, 3, 3, 3, 4, 7, 8, 8, 10, 10, 10, 12], [1, 3, 3, 3, 5, 5, 7, 7, 7, 8, 12, 12], [1, 3, 3, 3, 5, 5, 8, 8, 10, 10, 10, 12], [1, 3, 3, 3, 6, 7, 7, 7, 9, 9, 12, 12], [1, 3, 3, 3, 6, 7, 8, 8, 10, 10, 11, 12], [1, 3, 3, 3, 6, 8, 9, 9, 9, 10, 11, 12], [1, 3, 3, 3, 7, 7, 9, 10, 10, 10, 10, 12], [1, 3, 3, 3, 8, 9, 9, 9, 9, 10, 11, 12], [1, 3, 3, 4, 4, 4, 5, 8, 8, 9, 11, 12], [1, 3, 3, 4, 4, 4, 7, 9, 9, 10, 10, 12], [1, 3, 3, 4, 4, 8, 8, 9, 9, 10, 11, 12], [1, 3, 3, 4, 5, 5, 5, 5, 7, 7, 12, 12], [1, 3, 3, 4, 5, 5, 5, 7, 9, 10, 11, 12], [1, 3, 3, 4, 5, 5, 6, 6, 7, 9, 12, 12], [1, 3, 3, 4, 5, 5, 6, 7, 8, 9, 12, 12], [1, 3, 3, 4, 5, 6, 6, 6, 7, 11, 11, 12], [1, 3, 3, 4, 5, 7, 7, 7, 8, 10, 12, 12], [1, 3, 3, 4, 5, 7, 7, 8, 9, 11, 11, 12], [1, 3, 3, 4, 5, 8, 8, 9, 10, 10, 11, 12], [1, 3, 3, 4, 6, 8, 8, 8, 9, 10, 12, 12], [1, 3, 3, 4, 8, 8, 8, 9, 9, 10, 12, 12], [1, 3, 3, 5, 5, 5, 6, 6, 9, 9, 12, 12], [1, 3, 3, 5, 5, 5, 7, 7, 8, 10, 12, 12], [1, 3, 3, 5, 5, 5, 7, 9, 10, 10, 11, 12], [1, 3, 3, 5, 5, 6, 6, 6, 6, 10, 12, 12], [1, 3, 3, 5, 5, 6, 7, 8, 9, 10, 12, 12], [1, 3, 3, 5, 5, 7, 7, 7, 10, 11, 11, 12], [1, 3, 3, 5, 5, 7, 9, 9, 9, 11, 11, 12], [1, 3, 3, 5, 6, 6, 6, 7, 10, 11, 11, 12], [1, 3, 3, 5, 6, 7, 7, 7, 8, 11, 12, 12], [1, 3, 3, 5, 7, 9, 9, 9, 10, 11, 11, 12], [1, 3, 3, 5, 8, 8, 8, 8, 9, 11, 12, 12], [1, 3, 3, 5, 9, 9, 9, 9, 9, 10, 12, 12], [1, 3, 3, 6, 6, 6, 6, 7, 8, 11, 12, 12], [1, 3, 3, 6, 6, 7, 9, 9, 10, 10, 12, 12], [1, 3, 3, 6, 6, 8, 8, 8, 11, 11, 11, 12], [1, 3, 3, 6, 7, 8, 9, 9, 9, 11, 12, 12], [1, 3, 3, 7, 7, 8, 8, 9, 10, 11, 12, 12], [1, 3, 3, 7, 7, 9, 9, 9, 11, 11, 11, 12], [1, 3, 3, 8, 8, 8, 9, 9, 10, 11, 12, 12], [1, 3, 3, 9, 9, 9, 9, 9, 9, 11, 12, 12], [1, 3, 4, 4, 4, 4, 4, 4, 5, 5, 11, 12], [1, 3, 4, 4, 4, 4, 4, 4, 6, 10, 10, 12], [1, 3, 4, 4, 4, 4, 5, 6, 6, 10, 11, 12], [1, 3, 4, 4, 4, 4, 6, 6, 6, 8, 12, 12], [1, 3, 4, 4, 4, 4, 7, 9, 10, 10, 10, 12], [1, 3, 4, 4, 4, 6, 7, 7, 9, 11, 11, 12], [1, 3, 4, 4, 4, 8, 8, 8, 8, 10, 12, 12], [1, 3, 4, 4, 5, 5, 5, 5, 7, 9, 12, 12], [1, 3, 4, 4, 5, 5, 5, 7, 8, 11, 11, 12], [1, 3, 4, 4, 5, 5, 6, 7, 9, 11, 11, 12], [1, 3, 4, 4, 5, 5, 8, 9, 9, 9, 12, 12], [1, 3, 4, 4, 5, 6, 6, 7, 9, 10, 12, 12], [1, 3, 4, 4, 5, 6, 6, 9, 9, 11, 11, 12], [1, 3, 4, 4, 5, 6, 10, 10, 10, 10, 10, 12], [1, 3, 4, 4, 6, 9, 9, 9, 9, 10, 12, 12], [1, 3, 4, 4, 6, 10, 10, 10, 10, 10, 10, 12], [1, 3, 4, 4, 7, 7, 8, 10, 10, 11, 11, 12], [1, 3, 4, 5, 5, 5, 5, 5, 10, 10, 11, 12], [1, 3, 4, 5, 5, 5, 5, 6, 8, 10, 12, 12], [1, 3, 4, 5, 5, 5, 6, 6, 9, 10, 12, 12], [1, 3, 4, 5, 5, 5, 7, 9, 9, 10, 12, 12], [1, 3, 4, 5, 5, 7, 7, 8, 9, 11, 12, 12], [1, 3, 4, 5, 5, 7, 8, 8, 9, 11, 12, 12], [1, 3, 4, 5, 5, 7, 10, 10, 10, 10, 11, 12], [1, 3, 4, 5, 6, 6, 6, 6, 6, 11, 12, 12], [1, 3, 4, 5, 6, 6, 6, 7, 9, 11, 12, 12], [1, 3, 4, 5, 6, 6, 7, 7, 11, 11, 11, 12], [1, 3, 4, 5, 6, 6, 7, 8, 11, 11, 11, 12], [1, 3, 4, 5, 6, 7, 7, 9, 11, 11, 11, 12], [1, 3, 4, 5, 6, 7, 8, 10, 10, 10, 12, 12], [1, 3, 4, 5, 6, 9, 9, 10, 10, 11, 11, 12], [1, 3, 4, 5, 7, 8, 8, 9, 10, 11, 12, 12], [1, 3, 4, 5, 9, 9, 9, 9, 9, 11, 12, 12], [1, 3, 4, 6, 6, 7, 8, 10, 11, 11, 11, 12], [1, 3, 4, 6, 6, 8, 9, 9, 10, 11, 12, 12], [1, 3, 4, 6, 7, 8, 8, 8, 9, 12, 12, 12], [1, 3, 4, 6, 7, 9, 10, 10, 10, 10, 12, 12], [1, 3, 4, 6, 9, 9, 9, 10, 11, 11, 11, 12], [1, 3, 4, 7, 7, 8, 8, 9, 11, 11, 12, 12], [1, 3, 4, 7, 9, 9, 9, 10, 10, 11, 12, 12], [1, 3, 4, 8, 8, 8, 8, 8, 10, 12, 12, 12], [1, 3, 4, 8, 8, 8, 9, 11, 11, 11, 11, 12], [1, 3, 4, 8, 8, 9, 9, 9, 9, 12, 12, 12], [1, 3, 4, 9, 9, 9, 10, 10, 11, 11, 11, 12], [1, 3, 5, 5, 5, 5, 7, 9, 10, 10, 12, 12], [1, 3, 5, 5, 5, 5, 8, 9, 10, 10, 12, 12], [1, 3, 5, 5, 5, 6, 6, 10, 10, 11, 11, 12], [1, 3, 5, 5, 5, 8, 9, 10, 10, 10, 12, 12], [1, 3, 5, 5, 6, 6, 6, 7, 10, 11, 12, 12], [1, 3, 5, 5, 6, 6, 6, 8, 10, 11, 12, 12], [1, 3, 5, 5, 6, 6, 7, 9, 10, 11, 12, 12], [1, 3, 5, 5, 6, 6, 10, 10, 10, 11, 11, 12], [1, 3, 5, 5, 7, 8, 9, 10, 10, 11, 12, 12], [1, 3, 5, 6, 6, 6, 6, 7, 8, 12, 12, 12], [1, 3, 5, 6, 6, 6, 6, 8, 8, 12, 12, 12], [1, 3, 5, 6, 6, 6, 7, 8, 9, 12, 12, 12], [1, 3, 5, 6, 6, 6, 8, 10, 10, 11, 12, 12], [1, 3, 5, 6, 6, 7, 10, 10, 11, 11, 11, 12], [1, 3, 5, 6, 7, 7, 9, 11, 11, 11, 11, 12], [1, 3, 5, 6, 7, 8, 8, 9, 10, 12, 12, 12], [1, 3, 5, 7, 8, 8, 8, 10, 10, 12, 12, 12], [1, 3, 5, 7, 9, 9, 9, 10, 11, 11, 12, 12], [1, 3, 5, 10, 10, 10, 10, 10, 10, 10, 12, 12], [1, 3, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12], [1, 3, 6, 6, 6, 6, 6, 7, 11, 11, 12, 12], [1, 3, 6, 6, 6, 6, 8, 8, 10, 12, 12, 12], [1, 3, 6, 6, 6, 7, 8, 10, 11, 11, 12, 12], [1, 3, 6, 7, 8, 9, 9, 9, 11, 12, 12, 12], [1, 3, 6, 9, 9, 9, 9, 9, 11, 12, 12, 12], [1, 3, 6, 9, 9, 9, 9, 11, 11, 11, 12, 12], [1, 3, 6, 9, 10, 10, 10, 11, 11, 11, 11, 12], [1, 3, 7, 7, 8, 9, 10, 11, 11, 11, 12, 12], [1, 3, 7, 7, 8, 10, 11, 11, 11, 11, 11, 12], [1, 3, 8, 8, 9, 9, 10, 10, 11, 12, 12, 12], [1, 3, 8, 8, 10, 10, 10, 10, 10, 12, 12, 12], [1, 3, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12], [1, 4, 4, 4, 4, 4, 4, 6, 6, 6, 12, 12], [1, 4, 4, 4, 4, 4, 4, 6, 10, 10, 10, 12], [1, 4, 4, 4, 4, 4, 4, 8, 10, 10, 10, 12], [1, 4, 4, 4, 4, 4, 8, 10, 10, 10, 10, 12], [1, 4, 4, 4, 4, 5, 5, 5, 8, 9, 12, 12], [1, 4, 4, 4, 4, 5, 5, 6, 6, 11, 11, 12], [1, 4, 4, 4, 4, 5, 5, 6, 8, 11, 11, 12], [1, 4, 4, 4, 4, 5, 6, 9, 9, 9, 12, 12], [1, 4, 4, 4, 5, 5, 6, 8, 10, 11, 11, 12], [1, 4, 4, 4, 5, 6, 6, 8, 8, 11, 12, 12], [1, 4, 4, 4, 5, 8, 9, 9, 10, 10, 12, 12], [1, 4, 4, 4, 6, 6, 9, 10, 10, 11, 11, 12], [1, 4, 4, 4, 6, 7, 7, 7, 10, 11, 12, 12], [1, 4, 4, 4, 6, 8, 9, 9, 11, 11, 11, 12], [1, 4, 4, 4, 7, 8, 8, 10, 11, 11, 11, 12], [1, 4, 4, 4, 9, 9, 9, 10, 10, 10, 12, 12], [1, 4, 4, 5, 5, 6, 6, 9, 9, 11, 12, 12], [1, 4, 4, 5, 5, 6, 9, 9, 9, 11, 12, 12], [1, 4, 4, 5, 5, 8, 10, 10, 10, 11, 11, 12], [1, 4, 4, 5, 6, 8, 8, 10, 10, 11, 12, 12], [1, 4, 4, 5, 6, 9, 9, 10, 11, 11, 11, 12], [1, 4, 4, 5, 7, 7, 7, 7, 11, 11, 12, 12], [1, 4, 4, 5, 8, 8, 9, 9, 9, 12, 12, 12], [1, 4, 4, 5, 8, 9, 10, 10, 11, 11, 11, 12], [1, 4, 4, 6, 6, 8, 8, 8, 10, 12, 12, 12], [1, 4, 4, 6, 6, 8, 9, 9, 11, 11, 12, 12], [1, 4, 4, 6, 7, 10, 10, 10, 11, 11, 11, 12], [1, 4, 4, 6, 8, 8, 9, 10, 11, 11, 12, 12], [1, 4, 4, 6, 10, 10, 10, 10, 10, 10, 12, 12], [1, 4, 4, 7, 8, 8, 9, 10, 10, 12, 12, 12], [1, 4, 4, 8, 8, 9, 10, 10, 11, 11, 12, 12], [1, 4, 5, 5, 5, 5, 9, 9, 11, 11, 11, 12], [1, 4, 5, 5, 6, 7, 7, 7, 10, 12, 12, 12], [1, 4, 5, 5, 6, 9, 9, 9, 11, 11, 12, 12], [1, 4, 5, 6, 6, 10, 10, 10, 10, 11, 12, 12], [1, 4, 5, 6, 7, 10, 10, 11, 11, 11, 11, 12], [1, 4, 5, 7, 8, 8, 9, 10, 11, 12, 12, 12], [1, 4, 5, 8, 9, 9, 9, 10, 11, 12, 12, 12], [1, 4, 5, 8, 9, 9, 10, 11, 11, 11, 12, 12], [1, 4, 6, 6, 6, 8, 10, 10, 10, 12, 12, 12], [1, 4, 6, 6, 7, 8, 10, 11, 11, 11, 12, 12], [1, 4, 6, 6, 9, 9, 10, 11, 11, 11, 12, 12], [1, 4, 6, 6, 9, 10, 11, 11, 11, 11, 11, 12], [1, 4, 6, 7, 8, 8, 8, 9, 12, 12, 12, 12], [1, 4, 6, 8, 8, 9, 9, 9, 12, 12, 12, 12], [1, 4, 6, 8, 8, 9, 9, 11, 11, 12, 12, 12], [1, 4, 6, 8, 9, 11, 11, 11, 11, 11, 11, 12], [1, 4, 6, 10, 10, 10, 11, 11, 11, 11, 11, 12], [1, 4, 7, 7, 7, 10, 10, 10, 11, 12, 12, 12], [1, 4, 7, 9, 9, 9, 9, 10, 12, 12, 12, 12], [1, 4, 8, 8, 8, 8, 10, 10, 12, 12, 12, 12], [1, 4, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12], [1, 4, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12], [1, 5, 5, 5, 5, 5, 7, 7, 9, 12, 12, 12], [1, 5, 5, 5, 5, 6, 8, 11, 11, 11, 11, 12], [1, 5, 5, 5, 5, 7, 7, 9, 10, 12, 12, 12], [1, 5, 5, 5, 7, 7, 7, 9, 11, 12, 12, 12], [1, 5, 5, 5, 7, 8, 9, 11, 11, 11, 12, 12], [1, 5, 5, 5, 8, 9, 10, 10, 10, 12, 12, 12], [1, 5, 5, 5, 8, 9, 11, 11, 11, 11, 11, 12], [1, 5, 5, 6, 6, 6, 7, 7, 11, 12, 12, 12], [1, 5, 5, 6, 6, 7, 11, 11, 11, 11, 11, 12], [1, 5, 5, 6, 6, 10, 10, 10, 11, 11, 12, 12], [1, 5, 5, 7, 7, 7, 8, 8, 12, 12, 12, 12], [1, 5, 6, 6, 6, 8, 10, 10, 11, 12, 12, 12], [1, 5, 6, 6, 7, 9, 9, 9, 12, 12, 12, 12], [1, 5, 7, 7, 8, 9, 9, 11, 12, 12, 12, 12], [1, 5, 7, 7, 8, 10, 10, 10, 12, 12, 12, 12], [1, 5, 8, 9, 10, 10, 11, 11, 11, 12, 12, 12], [1, 6, 6, 6, 6, 8, 8, 10, 12, 12, 12, 12], [1, 6, 6, 7, 7, 9, 11, 11, 11, 12, 12, 12], [1, 6, 6, 10, 10, 11, 11, 11, 11, 11, 12, 12], [1, 6, 7, 9, 9, 9, 11, 11, 12, 12, 12, 12], [1, 6, 8, 8, 9, 10, 11, 11, 12, 12, 12, 12], [1, 6, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12], [1, 7, 7, 7, 7, 11, 11, 11, 11, 12, 12, 12], [1, 7, 8, 9, 10, 10, 10, 12, 12, 12, 12, 12], [1, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12]]

160117(2)

Ruby


フィボナッチ数列、トリボナッチ数列、テトラナッチ数列、…(3)

白石と黒石をn 個並べる。
先頭が黒石でk 個以上同じ色が続かない並べ方が何通りあるか考える。

求めるものをf(k, n) とおくと、

n < k のとき

f(k, n) = 2＾(k - 1)

n ≧ k のとき、

末尾が連続しないものが、f(k, n - 1) 通り,

末尾がちょうど2 連続同じ色のものが、f(k, n - 2) 通り,

末尾がちょうど3 連続同じ色のものが、f(k, n - 3) 通り,

…

末尾がちょうどk - 1 連続同じ色のものが、f(k, n - k - 1) 通り

あるので、

f(k, n) = f(k, n - 1) + … + f(k, n - k - 1)

f(k, n) の値を出力するコードは次のようになる。

# 先頭が黒石でk個以上同じ色が続かない
def f(k, n)
  ary = (1..k - 1).map{|i| 2 ** (i - 1)}
  (k..n).each{|i| ary[i - 1] = (1..k - 1).inject(0){|s, j| s += ary[i - 1 - j]}}
  ary[0..n - 1]
end

n = 20
(2..10).each{|k| p [k, f(k, n)]}

出力結果
[2, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]
[3, [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946]]
[4, [1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415]]
[5, [1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953]]
[6, [1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096]]
[7, [1, 2, 4, 8, 16, 32, 63, 125, 248, 492, 976, 1936, 3840, 7617, 15109, 29970, 59448, 117920, 233904, 463968]]
[8, [1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776]]
[9, [1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994]]
[10, [1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145]]

160117

Ruby

Lander, Parkin, and Selfridge conjecture

a, b, c, d, e を自然数とする。

a^4 + b^4 + c^4 + d^4 = e^4
を満たすとき、a, b, c, d のいずれかは315以上である。
また、
a^5 + b^5 + c^5 + d^5 = e^5
を満たすとき、a, b, c, d のいずれかは133以上である。

この二つのことを以下のコードで確認した。
（実行時間は10分弱位。）

n = 315
# i≦j≦k≦l≦n
(1..n).each{|i|
  (i..n).each{|j|
    (j..n).each{|k|
      (k..n).each{|l|
        z0 = i ** 4 + j ** 4 + k ** 4 + l ** 4
        z = (z0 ** 0.25).to_i
        p [i, j, k, l, z] if z0 == z ** 4
      }
    }
  }
}
n = 133
# i≦j≦k≦l≦n
(1..n).each{|i|
  (i..n).each{|j|
    (j..n).each{|k|
      (k..n).each{|l|
        z0 = i ** 5 + j ** 5 + k ** 5 + l ** 5
        z = (z0 ** 0.2).to_i
        p [i, j, k, l, z] if z0 == z ** 5
      }
    }
  }
}

出力結果
[30, 120, 272, 315, 353]
[27, 84, 110, 133, 144]

160114

Ruby

√5 の近似値の連分数展開(2)

√n の近似値または√n を連分数展開してみた。
（ただし、[a0; a1, … , an] を[a0, a1, … , an] と出力）

require 'bigdecimal'

# aが分子、bが分母
def f(a, b)
  return [a / b] if a % b == 0
  [a / b] + f(b, a % b)
end

d1, d2, n = 33, 30, 101
(1..n).each{|i|
  # √i の近似値
  str = BigDecimal.new(i).sqrt(d1).to_s("F")
  ary = str.scan(/\d+/).to_a
  # 小数点第d2位まで
  ary[1] = ary[1][0..d2 - 1] if ary[1].size > d2
  m = ary[1].size
  p [i, a = ary[0].to_i * 10 ** m + ary[1].to_i, b = 10 ** m]
  p f(a, b)
}

出力結果
[1, 10, 10]
[1]
[2, 1414213562373095048801688724209, 1000000000000000000000000000000]
[1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 41, 1, 1, 1, 1, 2, 3, 2, 1, 1, 7, 2, 3, 1, 3, 1, 1, 6, 1, 7, 1, 246, 3, 1, 1, 9, 138]
[3, 1732050807568877293527446341505, 1000000000000000000000000000000]
[1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 2, 1, 1, 12, 2, 29, 23, 1, 7, 4, 1, 3, 4, 88, 1, 3, 1, 12, 6, 10]
[4, 20, 10]
[2]
[5, 2236067977499789696409173668731, 1000000000000000000000000000000]
[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 1, 11, 9, 1, 10, 1, 27, 1, 2, 2, 1, 3, 3, 1, 4, 21, 4, 1, 1, 3, 1, 3, 1, 16, 1, 4, 2, 1, 1, 15]
[6, 2449489742783178098197284074705, 1000000000000000000000000000000]
[2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 3, 1, 6, 1, 2, 4, 1, 7, 1, 2, 1, 1, 4, 1, 7, 44, 2, 5, 1, 1, 3, 2, 10, 1, 7, 24, 1, 41]
[7, 2645751311064590590501615753639, 1000000000000000000000000000000]
[2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 7, 3, 1, 1, 2, 3, 32, 1, 1, 2, 18, 2, 1, 81, 89, 24, 8, 77]
[8, 2828427124746190097603377448419, 1000000000000000000000000000000]
[2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 3, 1, 1, 4, 1, 1197, 3, 1, 3, 4, 4, 19, 1, 3, 2, 10, 1, 12, 1, 2, 1, 1, 1, 1, 1, 1, 16, 8]
[9, 30, 10]
[3]
[10, 3162277660168379331998893544432, 1000000000000000000000000000000]
[3, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 1, 185, 2, 8, 1, 1, 1, 1, 24, 3, 2, 52, 1, 1, 2, 1, 1, 67, 2, 1, 1, 2, 1, 23]
[11, 3316624790355399849114932736670, 1000000000000000000000000000000]
[3, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 2, 4, 1, 85, 1, 7, 1, 2, 1, 6, 3, 1, 7, 5, 6, 1, 7, 1, 1, 1, 1, 1, 1, 1, 11, 1, 8, 13, 2]
[12, 3464101615137754587054892683011, 1000000000000000000000000000000]
[3, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 13, 18, 16, 1, 21, 1, 3, 1, 3, 1, 83, 1, 1, 3, 21, 2, 2, 5, 1, 4, 1, 9, 3, 1, 6]
[13, 3605551275463989293119221267470, 1000000000000000000000000000000]
[3, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 3, 8, 12, 2, 1, 10, 1, 34, 1, 3, 1, 12, 4, 5, 2, 1, 1, 1, 4, 2, 1, 2, 6, 8, 8, 2]
[14, 3741657386773941385583748732316, 1000000000000000000000000000000]
[3, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 6, 1, 1, 3, 1, 2, 1, 8, 2, 2, 1, 1, 39, 1, 2, 3, 1, 1, 12, 16, 2, 3, 3, 5, 5, 17, 18, 1, 2]
[15, 3872983346207416885179265399782, 1000000000000000000000000000000]
[3, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 5, 1, 5, 1, 3, 6, 2, 1, 1, 3, 7, 2, 5, 1, 7, 2, 2, 1, 1, 2, 1, 19, 3, 1, 2, 1, 1, 2, 2, 2, 4, 2, 1, 12, 2]
[16, 40, 10]
[4]
[17, 4123105625617660549821409855974, 1000000000000000000000000000000]
[4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 1, 5, 9, 5, 4, 1, 13, 2, 6, 1, 2, 195, 7, 12, 17, 2, 9, 1, 8, 1, 2]
[18, 4242640687119285146405066172629, 1000000000000000000000000000000]
[4, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 7, 1, 6, 2, 3, 3, 1, 1, 13, 3, 10, 3, 1, 1, 1, 1, 1, 4, 51, 1, 13, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 31]
[19, 4358898943540673552236981983859, 1000000000000000000000000000000]
[4, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 7, 9, 1, 5, 37, 2, 1, 2, 2, 2, 1, 8, 1, 1, 3, 1, 1, 7, 2, 6, 1, 6, 6, 8, 1, 4, 1, 1, 2, 1, 1, 11]
[20, 4472135954999579392818347337462, 1000000000000000000000000000000]
[4, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 7, 1, 5, 19, 1, 4, 1, 56, 2, 2, 7, 1, 1, 9, 10, 1, 1, 1, 1, 3, 1, 1, 8, 1, 7, 1, 9, 1, 3, 1, 7]
[21, 4582575694955840006588047193728, 1000000000000000000000000000000]
[4, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 2, 1, 3, 4, 1, 2, 1, 3, 1, 7, 1, 1, 4, 4, 1, 104, 1, 1, 3, 1, 1, 11]
[22, 4690415759823429554565630113544, 1000000000000000000000000000000]
[4, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 4, 1, 5, 2, 1, 25, 1, 8, 2, 71, 5, 9, 3, 1, 38, 1, 38, 1, 1, 1, 1, 3, 2, 1, 4]
[23, 4795831523312719541597438064162, 1000000000000000000000000000000]
[4, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 7, 1, 2, 6, 1, 2, 1, 2, 1, 77, 17, 1, 1, 1, 15, 2, 143, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 3, 2, 1, 1, 1, 1, 2]
[24, 4898979485566356196394568149411, 1000000000000000000000000000000]
[4, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 2, 1, 4, 1, 2, 1, 20, 1, 1, 1, 2, 28, 1, 2, 1, 1, 1, 1, 9, 1, 1, 1, 28, 1, 2, 3, 22, 1, 1, 1, 2, 1, 1, 1, 1, 6, 3]
[25, 50, 10]
[5]
[26, 5099019513592784830028224109022, 1000000000000000000000000000000]
[5, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 3, 1, 2, 3, 1, 2, 1, 1, 2, 1, 8, 1, 7, 1, 19, 1, 1, 1, 3, 2, 2, 5, 20, 1, 30, 1, 10, 1, 3, 2, 1, 5]
[27, 5196152422706631880582339024517, 1000000000000000000000000000000]
[5, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 6, 1, 16, 3, 3, 2, 1, 5, 6, 16, 4, 1, 5, 1, 1, 1, 8, 1, 2, 11, 9, 5, 1, 1, 3, 4, 2, 1, 4, 2]
[28, 5291502622129181181003231507278, 1000000000000000000000000000000]
[5, 3, 2, 3, 10, 3, 2, 3, 10, 3, 2, 3, 10, 3, 2, 3, 10, 3, 2, 3, 10, 3, 2, 3, 10, 3, 1, 14, 1, 1, 3, 1, 6, 16, 3, 2, 1, 8, 1, 2, 163, 44, 1, 1, 11, 1, 1, 3, 1, 1, 38]
[29, 5385164807134504031250710491540, 1000000000000000000000000000000]
[5, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 1, 3, 4, 20, 2, 1, 2, 3, 4, 1, 3, 1, 5, 3, 2, 16, 1, 9, 6, 1, 8, 2, 1, 1, 1, 9, 5]
[30, 5477225575051661134569697828008, 1000000000000000000000000000000]
[5, 2, 10, 2, 10, 2, 10, 2, 10, 2, 10, 2, 10, 2, 10, 2, 10, 2, 10, 2, 10, 2, 10, 2, 7, 1, 9, 1, 19, 2, 4, 1, 2, 5, 4, 2, 2, 3, 1, 9, 1, 4, 7, 7, 1, 2, 3, 2, 5, 1, 7]
[31, 5567764362830021922119471298918, 1000000000000000000000000000000]
[5, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, 5, 3, 1, 1, 10, 1, 1, 3, 2, 2, 1, 2, 44, 1, 12, 1, 7, 2, 1, 4, 1, 3, 1, 6, 1, 36, 1, 4, 4, 2, 2, 1, 2, 1, 2, 1, 1, 10, 1, 6]
[32, 5656854249492380195206754896838, 1000000000000000000000000000000]
[5, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 8, 10, 1, 598, 7, 1, 1, 8, 2, 39, 1, 1, 4, 5, 2, 6, 2, 1, 3, 4, 33, 4]
[33, 5744562646538028659850611468218, 1000000000000000000000000000000]
[5, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 1, 2, 1, 10, 91, 7, 3, 4, 167, 1, 1, 9, 1, 4, 1, 1, 29, 1, 5, 2, 2, 1, 1, 3, 5, 2, 2, 7]
[34, 5830951894845300470874152877545, 1000000000000000000000000000000]
[5, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 1, 1, 3, 1, 1, 1, 8, 1, 15, 1, 1, 2, 1, 1, 11, 1, 1, 1, 1, 5, 2, 1, 1, 1, 6, 1, 3, 2, 375, 1, 64]
[35, 5916079783099616042567328291561, 1000000000000000000000000000000]
[5, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 7, 2, 1, 1, 1, 2, 1, 1, 2, 11, 3, 7, 1, 5, 11, 4, 2, 10, 2, 2, 1, 3, 38, 2, 1, 32, 1, 4]
[36, 60, 10]
[6]
[37, 6082762530298219688999684245202, 1000000000000000000000000000000]
[6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 11, 1, 19, 11, 1, 1, 1, 1, 1, 1, 3, 5, 2, 2, 1, 4, 3, 2, 2, 19, 2, 1, 1, 34, 3, 1, 3, 1, 10, 2, 6]
[38, 6164414002968976450250192381454, 1000000000000000000000000000000]
[6, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 11, 1, 33, 30, 2, 2, 1, 1, 3, 8, 1, 1, 5, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 35, 2, 1, 1, 11, 2, 1, 13, 2]
[39, 6244997998398398205846893120939, 1000000000000000000000000000000]
[6, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 9, 1, 4, 1, 1, 3, 2, 4, 7, 1, 4, 1, 1, 9, 1, 1, 1, 108, 1, 1, 1, 1, 1, 33, 1, 1, 10, 4, 19, 12]
[40, 6324555320336758663997787088865, 1000000000000000000000000000000]
[6, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 12, 3, 5, 13, 1, 1, 2, 1, 15, 1, 3, 2, 1, 2, 3, 2, 1, 4, 1, 8, 1, 1, 8, 3, 2, 1, 75, 2, 1, 7, 1, 1, 3]
[41, 6403124237432848686488217674621, 1000000000000000000000000000000]
[6, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 1, 19, 1, 3, 1, 5, 1, 38, 1, 5, 3, 2, 38, 1, 1355, 1, 1, 1, 1, 1, 4, 3, 2, 7, 1, 7]
[42, 6480740698407860230965967436087, 1000000000000000000000000000000]
[6, 2, 12, 2, 12, 2, 12, 2, 12, 2, 12, 2, 12, 2, 12, 2, 12, 2, 12, 2, 12, 2, 5, 1, 116, 1, 1, 2, 6, 1, 2, 1, 2, 3, 6, 3, 1, 1, 5, 3, 1, 3, 1, 1, 2, 2, 5, 1, 10, 4, 6, 5, 1, 5]
[43, 6557438524302000652344109997636, 1000000000000000000000000000000]
[6, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 3, 1, 5, 1, 3, 1, 1, 12, 1, 1, 4, 1, 1, 1, 19, 3, 3, 4, 1, 1, 7, 5, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1036, 1, 6, 8, 2]
[44, 6633249580710799698229865473341, 1000000000000000000000000000000]
[6, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 4, 2, 5, 2, 5, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 7, 2, 3, 1, 1, 8, 1, 6, 1, 3, 5, 1, 2, 18, 1, 9]
[45, 6708203932499369089227521006193, 1000000000000000000000000000000]
[6, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 12, 1, 2, 2, 2, 1, 11, 1, 3, 29, 1, 2, 1, 85, 9, 3, 2, 1, 1, 1, 63, 1, 1, 1, 12, 2, 1, 1, 5, 3, 1, 2, 6, 1, 4, 3]
[46, 6782329983125268139064556326625, 1000000000000000000000000000000]
[6, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12, 1, 3, 2, 1, 24, 1, 19, 2, 8, 9, 2, 7, 1, 1, 6, 5, 1, 5, 1, 1, 3, 1, 1, 1, 303]
[47, 6855654600401044124935871449084, 1000000000000000000000000000000]
[6, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 2, 2, 1, 1, 9, 1, 2, 12, 3, 2, 2, 3, 5, 85, 1, 6, 1, 10, 1, 1, 3, 2, 3, 1, 3, 2, 9, 1, 4]
[48, 6928203230275509174109785366023, 1000000000000000000000000000000]
[6, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 12, 1, 1, 1, 35, 2, 1, 2, 2, 2, 1, 2, 3, 4, 1, 404, 1, 5, 1, 2, 4, 1, 21, 4, 1, 1, 2, 11, 1, 6, 2]
[49, 70, 10]
[7]
[50, 7071067811865475244008443621048, 1000000000000000000000000000000]
[7, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 2, 3, 4, 1, 13, 1, 1, 36, 6932, 1, 2, 4, 1, 2, 1, 6, 1, 1, 8, 2, 1, 1, 13]
[51, 7141428428542849997999399811367, 1000000000000000000000000000000]
[7, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 4, 1, 6, 3, 1, 2, 3, 3, 3, 2, 1, 2, 5, 10, 1, 5, 2, 4, 1, 3, 1, 11, 2, 2, 3, 19, 1, 1, 2, 1, 1, 1, 1, 1, 2]
[52, 7211102550927978586238442534940, 1000000000000000000000000000000]
[7, 4, 1, 2, 1, 4, 14, 4, 1, 2, 1, 4, 14, 4, 1, 2, 1, 4, 14, 4, 1, 2, 1, 4, 14, 4, 1, 2, 1, 4, 7, 4, 24, 1, 2, 5, 2, 17, 2, 1, 1, 25, 2, 10, 1, 3, 9, 1, 2, 1, 12, 4, 17]
[53, 7280109889280518271097302491527, 1000000000000000000000000000000]
[7, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 3, 1, 1, 3, 14, 9, 1, 1, 5, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 6, 2, 3, 1, 1, 1, 4, 4, 1, 5, 2, 1, 2, 1, 1, 1, 3, 7, 3, 2, 10, 1, 2, 2]
[54, 7348469228349534294591852224117, 1000000000000000000000000000000]
[7, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 14, 2, 1, 6, 1, 2, 13, 1, 6, 1, 1, 4, 3, 2, 2, 2, 2, 4, 9, 1, 1, 64, 2, 1, 6, 6, 1, 2, 2, 4, 1, 3, 2, 2, 1, 1, 4, 1, 1, 2, 4]
[55, 7416198487095662948711397440800, 1000000000000000000000000000000]
[7, 2, 2, 2, 14, 2, 2, 2, 14, 2, 2, 2, 14, 2, 2, 2, 14, 2, 2, 2, 14, 2, 2, 2, 14, 2, 2, 2, 5, 2, 2, 1, 2, 1, 2, 1, 3, 4, 1, 7, 1, 1, 3, 1, 6, 1, 6, 11, 6, 22, 3, 1, 2, 4]
[56, 7483314773547882771167497464633, 1000000000000000000000000000000]
[7, 2, 14, 2, 14, 2, 14, 2, 14, 2, 14, 2, 14, 2, 14, 2, 14, 2, 14, 2, 14, 2, 4, 2, 10, 4, 4, 1, 1, 69, 2, 12, 1, 3, 4, 27, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 5, 2, 4, 1, 2]
[57, 7549834435270749697236684806946, 1000000000000000000000000000000]
[7, 1, 1, 4, 1, 1, 14, 1, 1, 4, 1, 1, 14, 1, 1, 4, 1, 1, 14, 1, 1, 4, 1, 1, 14, 1, 1, 4, 1, 1, 14, 1, 1, 4, 1, 1, 14, 2, 14, 108, 3, 1, 1, 1, 1, 1, 12, 1, 7, 1, 131, 1, 1, 9, 1, 1, 1, 67, 4, 1, 1, 5, 2]
[58, 7615773105863908285661411027158, 1000000000000000000000000000000]
[7, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 2, 7, 6, 3, 17, 1, 1, 3, 4, 1, 4, 3, 45, 3, 1, 1, 2, 1, 5, 1, 2, 1, 15, 1, 4, 2, 11, 7]
[59, 7681145747868608175769687021731, 1000000000000000000000000000000]
[7, 1, 2, 7, 2, 1, 14, 1, 2, 7, 2, 1, 14, 1, 2, 7, 2, 1, 14, 1, 2, 7, 2, 1, 14, 1, 2, 7, 2, 1, 14, 23, 1, 7, 1, 2, 2, 10, 1, 3, 1, 1, 14, 1, 3, 1, 1, 2, 7, 1, 1, 1, 1, 5, 2, 2, 1, 16, 2, 6, 1, 1, 18]
[60, 7745966692414833770358530799564, 1000000000000000000000000000000]
[7, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 2, 2, 1, 7, 3, 5, 7, 3, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 9, 1, 1, 1, 2, 1, 3, 2, 2, 2, 1, 1, 1, 2, 26]
[61, 7810249675906654394129722735759, 1000000000000000000000000000000]
[7, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14, 1, 4, 2, 1, 1, 14, 6, 5, 5, 3, 1, 3, 39, 1, 2, 1, 4, 1, 1, 45, 2, 1, 2, 1, 7, 1, 1, 1, 109, 2]
[62, 7874007874011811019685034448812, 1000000000000000000000000000000]
[7, 1, 6, 1, 14, 1, 6, 1, 14, 1, 6, 1, 14, 1, 6, 1, 14, 1, 6, 1, 14, 1, 6, 1, 14, 1, 6, 1, 14, 1, 6, 1, 4, 2, 1, 5, 5, 4, 3, 1, 1, 1, 227, 2, 1, 5, 3, 1, 1, 1, 18, 2, 1, 1, 5, 1, 1, 1, 1, 6, 3]
[63, 7937253933193771771504847260917, 1000000000000000000000000000000]
[7, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 14, 1, 7, 2, 2, 2, 1, 7, 1, 97, 1, 3, 1, 5, 2, 4, 27, 267, 8, 24, 25, 1, 2]
[64, 80, 10]
[8]
[65, 8062257748298549652366613230303, 1000000000000000000000000000000]
[8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 15, 1, 219, 1, 5, 1, 1, 4, 4, 2, 4, 7, 1, 5, 4, 65, 1, 4, 3, 18, 7, 1, 2, 1, 1, 1, 18]
[66, 8124038404635960360459883568266, 1000000000000000000000000000000]
[8, 8, 16, 8, 16, 8, 16, 8, 16, 8, 16, 8, 16, 8, 16, 9, 4, 9, 1, 6, 1, 3, 13, 1, 1, 2, 14, 7, 2, 23, 1, 2, 9, 8, 1, 2, 1, 11, 1, 14]
[67, 8185352771872449969953703724733, 1000000000000000000000000000000]
[8, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 16, 5, 2, 1, 1, 7, 1, 1, 2, 5, 15, 2, 2, 2, 7, 3, 1, 2, 6, 35, 1, 2, 370, 1, 2, 4, 2, 312, 1, 20, 1, 6, 3]
[68, 8246211251235321099642819711948, 1000000000000000000000000000000]
[8, 4, 16, 4, 16, 4, 16, 4, 16, 4, 16, 4, 16, 4, 16, 4, 16, 4, 2, 2, 1, 1, 4, 10, 2, 2, 6, 1, 2, 1, 2, 1, 5, 97, 1, 1, 3, 24, 8, 1, 2, 1, 4, 2, 4, 3]
[69, 8306623862918074852584262744907, 1000000000000000000000000000000]
[8, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 3, 3, 16, 3, 3, 1, 4, 1, 2, 1, 2, 1, 15, 8, 1, 21, 3, 1, 13, 1, 3, 1, 1, 49, 4, 4, 18, 11, 9, 5, 1, 15]
[70, 8366600265340755479781720257851, 1000000000000000000000000000000]
[8, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 3, 3, 1, 9, 1, 2, 5, 47, 4, 4, 1, 6, 3, 4, 6, 59, 1, 2, 6, 30, 4, 6, 2]
[71, 8426149773176358630634139906202, 1000000000000000000000000000000]
[8, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 16, 2, 2, 1, 7, 1, 2, 2, 13, 6, 3, 2, 3, 1, 1, 8, 1, 13, 2, 23, 1, 3, 1, 1, 1, 24, 4, 11, 1, 3, 1, 3, 4, 9, 9]
[72, 8485281374238570292810132345258, 1000000000000000000000000000000]
[8, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 16, 2, 15, 1, 2, 1, 2, 1, 1, 7, 27, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 1, 25, 2, 6, 1, 4, 3, 1, 2, 1, 2, 1, 1, 2, 1, 15]
[73, 8544003745317531167871648326239, 1000000000000000000000000000000]
[8, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 5, 1, 1, 16, 1, 1, 5, 4, 1, 1, 10, 1, 3, 1, 1, 4, 2, 1, 20, 1, 1, 1, 3, 1, 4, 1, 1, 7, 1, 3, 1, 3, 15, 143, 4, 71]
[74, 8602325267042626771729473535049, 1000000000000000000000000000000]
[8, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 16, 1, 1, 1, 1, 11, 1, 14, 1, 1, 1, 1, 3, 5, 2, 5, 2, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 31, 2, 1, 17, 2, 1, 5, 1, 8, 1, 3, 5]
[75, 8660254037844386467637231707529, 1000000000000000000000000000000]
[8, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 14, 3, 1, 3, 24, 3, 1, 1, 33, 12, 1, 1, 1, 1, 7, 3, 1, 3, 1, 4, 2, 1, 38, 2, 6, 1, 5, 2, 4]
[76, 8717797887081347104473963967719, 1000000000000000000000000000000]
[8, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16, 5, 2, 8, 4, 2, 1, 1, 3, 1, 4, 5, 11, 5, 3, 1, 11, 1, 6, 1, 4, 1, 1, 1, 2, 49, 1, 9, 2, 2]
[77, 8774964387392122060406388307416, 1000000000000000000000000000000]
[8, 1, 3, 2, 3, 1, 16, 1, 3, 2, 3, 1, 16, 1, 3, 2, 3, 1, 16, 1, 3, 2, 3, 1, 16, 1, 3, 2, 3, 1, 16, 1, 3, 3, 1, 1, 1, 30, 1, 19, 1, 13, 1, 7, 1, 13, 1, 1, 38, 17, 3, 43, 1, 8, 1, 6]
[78, 8831760866327846854764042726959, 1000000000000000000000000000000]
[8, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 2, 1, 31, 3, 8, 4, 1, 1, 3, 1, 6, 1, 16, 9, 1, 19, 4, 4, 2, 3, 1, 50, 1, 5, 6]
[79, 8888194417315588850091441675408, 1000000000000000000000000000000]
[8, 1, 7, 1, 16, 1, 7, 1, 16, 1, 7, 1, 16, 1, 7, 1, 16, 1, 7, 1, 16, 1, 7, 1, 16, 1, 7, 1, 12, 1, 5, 3, 1, 7, 2, 8, 3, 1, 4, 1, 1, 5, 1, 7, 1, 2, 2, 1, 5, 2, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 5]
[80, 8944271909999158785636694674925, 1000000000000000000000000000000]
[8, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 4, 2, 2, 41, 9, 3, 1, 1, 2, 4, 1, 1, 5, 4, 2, 1, 2, 2, 2, 1, 14, 2, 4, 88]
[81, 90, 10]
[9]
[82, 9055385138137416626573808166984, 1000000000000000000000000000000]
[9, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 17, 1, 24, 7, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 11, 1, 45, 1, 1, 2, 2, 79, 2, 3, 33, 3, 1, 2]
[83, 9110433579144298881945626104688, 1000000000000000000000000000000]
[9, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 18, 9, 13, 6, 1, 1, 9, 1, 3, 7, 2, 11, 6, 1, 1, 1, 1, 1, 10, 1, 4, 1, 2, 4, 1, 3, 2, 2, 1, 2, 12, 5]
[84, 9165151389911680013176094387456, 1000000000000000000000000000000]
[9, 6, 18, 6, 18, 6, 18, 6, 18, 6, 18, 6, 18, 6, 18, 6, 12, 1, 2, 3, 4, 2, 7, 2, 2, 1, 2, 1, 1, 16, 9, 2, 2, 52, 3, 1, 1, 3, 1, 5]
[85, 9219544457292887310002274281762, 1000000000000000000000000000000]
[9, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 3, 1, 2, 1, 2, 2, 74, 2, 1, 1, 1, 1, 8, 1, 1, 1, 44, 2, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 3, 5, 7, 3, 1, 3, 2]
[86, 9273618495495703752516416073990, 1000000000000000000000000000000]
[9, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 1, 1, 3, 18, 3, 1, 1, 1, 8, 1, 3, 1, 2, 4, 1, 1, 12, 1, 1, 2, 1, 8, 2, 1, 1, 1, 2, 9, 1, 1, 21, 1, 17, 1, 1, 1, 6, 5, 17, 3]
[87, 9327379053088815045554475542320, 1000000000000000000000000000000]
[9, 3, 18, 3, 18, 3, 18, 3, 18, 3, 18, 3, 18, 3, 18, 3, 18, 3, 9, 1, 1, 1, 4, 4, 1, 2, 5, 5, 1, 1, 1, 3, 6, 41, 2, 4, 78, 111, 1, 2]
[88, 9380831519646859109131260227088, 1000000000000000000000000000000]
[9, 2, 1, 1, 1, 2, 18, 2, 1, 1, 1, 2, 18, 2, 1, 1, 1, 2, 18, 2, 1, 1, 1, 2, 18, 2, 1, 1, 1, 2, 18, 2, 1, 1, 1, 2, 10, 1, 2, 5, 1, 12, 2, 4, 4, 35, 1, 1, 2, 18, 1, 1, 78, 1, 18, 1, 4, 7, 1, 2, 2]
[89, 9433981132056603811320660377622, 1000000000000000000000000000000]
[9, 2, 3, 3, 2, 18, 2, 3, 3, 2, 18, 2, 3, 3, 2, 18, 2, 3, 3, 2, 18, 2, 3, 3, 2, 19, 10, 3, 1, 1, 38, 12, 1, 7, 2, 2, 265, 3, 20, 1, 2, 3, 6, 3, 9, 3, 3]
[90, 9486832980505137995996680633298, 1000000000000000000000000000000]
[9, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 13, 1, 9, 34, 1, 16, 1, 3, 1, 2, 1, 3, 1, 8, 2, 36, 2, 1, 2, 1, 1, 18, 3, 2, 1, 1, 1, 1, 3, 1, 3]
[91, 9539392014169456491526215860232, 1000000000000000000000000000000]
[9, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 5, 1, 5, 1, 1, 18, 1, 1, 6, 3, 2, 11, 2, 1, 9, 3, 1, 405, 1, 3, 2, 1, 1, 60, 1, 70, 2, 4, 1, 3, 1, 1, 2]
[92, 9591663046625439083194876128325, 1000000000000000000000000000000]
[9, 1, 1, 2, 4, 2, 1, 1, 18, 1, 1, 2, 4, 2, 1, 1, 18, 1, 1, 2, 4, 2, 1, 1, 18, 1, 1, 2, 4, 2, 1, 1, 18, 1, 1, 2, 4, 4, 2, 1, 1, 1, 1, 4, 13, 1, 5, 1, 3, 3, 8, 1, 4, 2, 6, 16, 1, 8, 3, 3, 2, 3, 23]
[93, 9643650760992954995760031047432, 1000000000000000000000000000000]
[9, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 6, 4, 1, 1, 1, 18, 1, 1, 1, 4, 7, 1, 1, 4, 2, 1, 2, 126, 2, 2, 3, 5, 5, 1, 1, 20, 1, 5, 3, 2, 4, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3]
[94, 9695359714832658028148881150845, 1000000000000000000000000000000]
[9, 1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 18, 1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 18, 1, 2, 3, 1, 1, 4, 5, 1, 1, 12, 1, 2, 3, 1, 2, 1, 7, 2, 1, 4, 9, 7, 1, 8, 1, 1, 4, 10, 6, 2, 1, 1, 9, 6]
[95, 9746794344808963906838413199899, 1000000000000000000000000000000]
[9, 1, 2, 1, 18, 1, 2, 1, 18, 1, 2, 1, 18, 1, 2, 1, 18, 1, 2, 1, 18, 1, 2, 1, 18, 1, 2, 1, 18, 1, 2, 1, 17, 1, 2, 6, 36, 14, 2, 3, 1, 1, 1, 25, 4, 2, 1, 2, 1, 47, 1, 1, 1, 8, 3, 17, 1, 16, 2]
[96, 9797958971132712392789136298823, 1000000000000000000000000000000]
[9, 1, 3, 1, 18, 1, 3, 1, 18, 1, 3, 1, 18, 1, 3, 1, 18, 1, 3, 1, 18, 1, 3, 1, 18, 1, 3, 1, 18, 1, 3, 1, 5, 1, 1, 4, 4, 1, 1, 2, 1, 1, 28, 1, 3, 2, 2, 2, 3, 3, 2, 1, 3, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 3, 14, 1, 1, 3, 3, 1, 1, 2]
[97, 9848857801796104721746211414917, 1000000000000000000000000000000]
[9, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18, 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 1, 2, 1, 2, 1, 3, 4, 8, 3, 4, 1, 1, 1, 16, 1, 1, 2, 1, 1, 2, 2, 3, 4, 1, 9, 3, 1, 26, 1, 12, 25]
[98, 9899494936611665341611821069467, 1000000000000000000000000000000]
[9, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 1, 1, 3, 1, 2, 6, 282, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 3, 5, 1, 8, 3, 1, 1, 2, 1, 8, 3]
[99, 9949874371066199547344798210012, 1000000000000000000000000000000]
[9, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 1, 18, 22, 1, 1, 21, 1, 3, 8, 4, 6, 1, 3, 1, 1, 5, 2, 2, 2, 5, 1, 2, 4, 1, 1, 1, 4, 2, 2, 15]
[100, 100, 10]
[10]
[101, 10049875621120890270219264912759, 1000000000000000000000000000000]
[10, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 13, 3, 1, 9, 3, 3, 4, 2, 5, 4, 33, 6, 2, 3, 1, 66, 2, 16, 1, 1, 4, 1, 4, 1, 2, 1, 3]

160113

Ruby

√5 の近似値の連分数展開(1)

√5 の近似値を連分数展開してみた。
（ただし、[a0; a1, … , an] を[a0, a1, … , an] と出力）
なお、√5 の連分数展開は、
オンライン整数列大辞典の
A040002（http://oeis.org/A040002/list）
に載っています。

# aが分子、bが分母
def f(a, b)
  return [a / b] if a % b == 0
  [a / b] + f(b, a % b)
end

def g(ary)
  # √5 = 2.23…
  n, d = ary.join.to_i, 10 ** (ary.size - 1)
  [f(n, d), n]
end

# OEIS A002163のデータ
ary =
[2,2,3,6,0,6,7,9,7,7,4,9,9,7,8,9,6,9,6,4,0,9,1,7,
 3,6,6,8,7,3,1,2,7,6,2,3,5,4,4,0,6,1,8,3,5,9,6,1,1,
 5,2,5,7,2,4,2,7,0,8,9,7,2,4,5,4,1,0,5,2,0,9,2,5,6,
 3,7,8,0,4,8,9,9,4,1,4,4,1,4,4,0,8,3,7,8,7,8,2,2,7]

(0..98).each{|i| p g(ary[0..i])}

出力結果
[[2], 2]
[[2, 5], 22]
[[2, 4, 2, 1, 7], 223]
[[2, 4, 4, 4, 1, 2], 2236]
[[2, 4, 4, 4, 1, 2], 22360]
[[2, 4, 4, 4, 3, 1, 1, 6, 14], 223606]
[[2, 4, 4, 4, 4, 7, 3, 3, 8, 2, 2], 2236067]
[[2, 4, 4, 4, 4, 4, 2, 1, 1, 2, 12, 1, 12, 1, 2], 22360679]
[[2, 4, 4, 4, 4, 4, 3, 1, 960, 1, 8, 2], 223606797]
[[2, 4, 4, 4, 4, 4, 4, 4, 1, 1, 3, 1, 7, 1, 119, 1, 3], 2236067977]
[[2, 4, 4, 4, 4, 4, 4, 4, 3, 2, 1, 2, 3, 2, 2, 5, 73], 22360679774]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 7, 1, 1, 51, 1, 103, 2, 2, 2], 223606797749]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 5, 1, 4, 1, 1, 1, 5, 1, 1, 2, 25, 2, 2], 2236067977499]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 1, 17, 2, 3, 7, 1, 2, 2, 1, 2, 1, 3, 4, 4, 2], 22360679774997]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 13, 1, 2, 1, 1, 1, 1, 8, 1, 1, 2, 2, 20, 2, 4], 223606797749978]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 1, 1, 2, 1, 5, 1, 1, 2, 10, 1, 4, 5, 1, 1, 2, 2, 5, 2, 3], 2236067977499789]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 7, 1, 1, 1, 2, 19, 3, 1, 3, 1, 560, 1, 2], 22360679774997896]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 2, 1, 1, 6, 1, 1, 1, 1, 2, 1, 6, 5, 1, 3, 1, 2, 4, 1, 2, 1, 11, 2], 223606797749978969]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 10, 1, 1, 1, 6, 2, 2, 2, 1, 7, 2, 5, 1, 1, 1, 2, 4], 2236067977499789696]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 1, 1, 1, 3, 5, 6, 2, 10, 2, 26, 8, 3, 4, 2], 22360679774997896964]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 1, 1, 1, 3, 5, 6, 2, 10, 2, 26, 8, 3, 4, 2], 223606797749978969640]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 2, 5, 3, 20, 3, 1, 1, 7, 3, 7, 1, 1, 1, 1, 8, 9, 1, 1, 1, 1, 2], 2236067977499789696409]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 6, 2, 1, 19, 2, 3, 3, 2, 4, 2, 2, 20, 1, 5, 1, 1, 1, 1, 2, 1, 1, 8, 2], 22360679774997896964091]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 9, 1, 1, 1, 7, 2, 1, 2, 1, 18, 1, 5, 14, 2, 2, 7, 2, 24, 1, 6, 1, 1, 1, 2], 223606797749978969640917]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 2, 1, 2, 1, 2, 1, 12, 10, 1, 1, 2, 7, 1, 1, 1, 14, 2, 1, 1, 8, 1, 3, 1, 1, 5, 6, 1, 1, 2], 2236067977499789696409173]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 1, 1, 5, 1, 1, 1, 8, 1, 73, 1, 1, 11, 8, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 34, 1, 6], 22360679774997896964091736]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 7, 2, 8, 4, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 4, 1, 15, 1, 1, 20, 1, 5, 1, 3, 5, 3, 2, 1, 2, 2], 223606797749978969640917366]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 40, 2, 1, 42, 1, 2, 1, 3, 2, 83, 1, 4, 1, 1, 1, 1, 24, 7, 2, 2, 8], 2236067977499789696409173668]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 2, 3, 3, 1, 2, 1, 4, 1, 2, 1, 1, 5, 1, 9, 4, 1, 3, 1, 2, 107, 1, 1, 1, 2, 1, 7, 4, 1, 21], 22360679774997896964091736687]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 15, 4, 40, 2, 2, 5, 1, 2, 18, 1, 1, 73, 3, 2, 18, 2, 65, 1, 10], 223606797749978969640917366873]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 1, 11, 9, 1, 10, 1, 27, 1, 2, 2, 1, 3, 3, 1, 4, 21, 4, 1, 1, 3, 1, 3, 1, 16, 1, 4, 2, 1, 1, 15], 2236067977499789696409173668731]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 1, 42, 1, 1, 1, 168, 4, 1, 6, 1, 3, 8, 1, 22, 1, 14, 1, 7, 2, 1, 6, 1, 1, 19], 22360679774997896964091736687312]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 2, 1, 4, 1, 3, 1, 1, 2, 1, 9, 2, 1, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 11, 2, 1, 5, 1, 1, 1, 2, 1, 1, 2, 2, 5, 2, 4, 49, 2], 223606797749978969640917366873127]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 1, 20, 11, 1, 1, 11, 2, 2, 1, 5, 12, 1, 2, 1, 4, 2, 10, 2, 1, 1, 1, 62, 19, 2, 1, 5, 8], 2236067977499789696409173668731276]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 42, 1, 1, 2, 31, 3, 1, 4, 1, 1, 1, 1, 32, 15, 13, 3, 1, 3, 1, 5, 1, 8, 5, 1, 3, 6, 6, 1, 4], 22360679774997896964091736687312762]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 5, 2, 11, 6, 2, 141, 8, 2, 4, 3, 1, 94, 1, 23, 1, 6, 1, 5, 2, 4], 223606797749978969640917366873127623]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 1, 1, 4, 14, 1, 6, 1, 1, 1, 18, 1, 7, 39, 2, 2, 24, 1, 2, 1, 4, 2, 4, 39, 18, 1, 3, 3, 1, 1, 4, 2], 2236067977499789696409173668731276235]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 80, 1, 1, 1, 1, 2, 2, 1, 3, 1, 52, 1, 1, 1, 2, 22, 1, 1, 1, 3, 2, 3, 5, 13, 4, 2, 1, 3, 3, 1, 1, 1, 2, 1, 1, 13, 22], 22360679774997896964091736687312762354]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 2, 1, 65, 11, 1, 3, 9, 1, 1, 21, 3, 2, 1, 3, 1, 27, 4, 3, 1, 1, 42, 47, 6, 2, 1, 12, 4], 223606797749978969640917366873127623544]
[[2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 2, 1, 65, 11, 1, 3, 9, 1, 1, 21, 3, 2, 1, 3, 1, 27, 4, 3, 1, 1, 42, 47, 6, 2, 1, 12, 4], 2236067977499789696409173668731276235440]
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