160628(2)

Ruby

Numbers n such that n^2 divides A000073(n)(1)

オンライン整数列大辞典のA274518 について、
このような性質をもつ数を見つけたので、
検証してみる。

require 'matrix'
def power(a, n, mod)
  return Matrix.I(a.row_size) if n == 0
  m = power(a, n >> 1, mod)
  m = (m * m).map{|i| i % mod}
  return m if n & 1 == 0
  (m * a).map{|i| i % mod}
end
def f(m, n, mod)
  ary0 = Array.new(m, 0)
  ary0[0] = 1
  v = Vector.elements(ary0)
  ary1 = [Array.new(m, 1)]
  (0..m - 2).each{|i|
    ary2 = Array.new(m, 0)
    ary2[i] = 1
    ary1 << ary2
  }
  a = Matrix[*ary1]
  (power(a, n, mod) * v)[m - 1]
end

# チェック用
ary0 = (1..103).to_a
p ary0.select{|i| f(3, i, i * i) == 0}

ary1 = [1, 103, 112, 2621, 30576, 77168, 694512, 9919728, 24770928, 55638128, 57268848, 80995824, 1300820976, 2346310512, 151508024304, 151705178352, 393783442416]
p ary1.select{|i| f(3, i, i * i) == 0}

出力結果
[1, 103]
[1, 103, 112, 2621, 30576, 77168, 694512, 9919728, 24770928, 55638128, 57268848, 80995824, 1300820976, 2346310512, 151508024304, 151705178352, 393783442416]