コラッツの問題でｎが与えられたとき、そのコラッツの系列のなかに何個素数があるかを計算してみた。一種の素数競争であります。
例を81で示す。このコラッツ系列は{81, 122, 61, 92, 46, 23, 35, 53, 80, 40, 20, 10, 5, 8, 4, 2, 1}であります。そのなかの素数は{61, 23, 53, 5, 2}であります。つまりは「５」個の素数でありますな。
こんな感じの操作を飽くことなく繰り返すわけです。
最初の１０００個の自然数でのコラッツの系列での素数の数であります。
{0, 1, 3, 1, 2, 3, 6, 1, 6, 2, 5, 3, 3, 6, 4, 1, 4, 6, 7, 2, 1, 5, 4, 3, 7, 3, 25, 6, 6, 4, 24, 1, 7, 4, 3, 6, 7, 7, 11, 2, 25, 1, 8, 5, 4, 4, 23, 3, 7, 7, 6, 3, 3, 25, 24, 6, 8, 6, 11, 4, 5, 24, 20, 1, 7, 7, 9, 4, 3, 3, 22, 6, 25, 7, 2, 7, 6, 11, 11, 2, 5, 25, 24, 1, 1, 8, 9, 5, 10, 4, 20, 4, 3, 23, 20, 3, 26, 7, 7, 7, 8, 6, 19, 3, 11, 3, 21, 25, 26, 24, 15, 6, 2, 8, 8, 6, 5, 11, 10, 4, 20, 5, 13, 24, 23, 20, 12, 1, 26, 7, 9, 7, 7, 9, 9, 4, 20, 3, 13, 3, 3, 22, 20, 6, 26, 25, 24, 7, 7, 2, 3, 7, 8, 6, 18, 11, 12, 11, 13, 2, 20, 5, 5, 25, 24, 24, 15, 1, 12, 1, 28, 8, 8, 9, 17, 5, 7, 10, 10, 4, 5, 20, 18, 4, 13, 3, 13, 23, 22, 20, 11, 3, 27, 26, 25, 7, 8, 7, 23, 7, 3, 8, 9, 6, 6, 19, 20, 3, 12, 11, 12, 3, 2, 21, 20, 25, 5, 26, 13, 24, 24, 15, 14, 6, 12, 2, 2, 8, 9, 8, 29, 6, 18, 5, 24, 11, 10, 10, 13, 4, 6, 20, 20, 5, 4, 13, 13, 24, 13, 23, 14, 20, 20, 12, 9, 1, 28, 26, 26, 7, 7, 9, 17, 7, 23, 7, 3, 9, 9, 9, 9, 4, 6, 20, 18, 3, 4, 13, 12, 3, 13, 3, 13, 22, 21, 20, 10, 6, 5, 26, 25, 25, 25, 24, 15, 7, 14, 7, 22, 2, 2, 3, 9, 7, 9, 8, 10, 6, 6, 18, 20, 11, 25, 12, 7, 11, 11, 13, 12, 2, 6, 20, 20, 5, 5, 5, 26, 25, 13, 24, 5, 24, 23, 15, 13, 1, 21, 12, 12, 1, 1, 28, 27, 8, 26, 8, 29, 9, 10, 17, 16, 5, 24, 7, 7, 10, 9, 10, 12, 4, 9, 5, 11, 20, 20, 18, 11, 4, 4, 13, 13, 3, 4, 13, 12, 23, 13, 22, 12, 20, 20, 11, 9, 3, 5, 27, 26, 26, 26, 25, 27, 7, 15, 8, 16, 7, 8, 23, 21, 7, 3, 3, 2, 8, 7, 9, 8, 6, 11, 6, 29, 19, 18, 20, 25, 3, 25, 12, 12, 11, 12, 12, 4, 3, 12, 2, 10, 21, 20, 20, 10, 25, 6, 5, 5, 26, 25, 13, 15, 24, 5, 24, 15, 15, 15, 14, 18, 6, 22, 12, 13, 2, 1, 2, 9, 8, 28, 9, 26, 8, 9, 29, 28, 6, 10, 18, 19, 5, 5, 24, 24, 11, 7, 10, 2, 10, 10, 13, 12, 4, 9, 6, 4, 20, 20, 20, 30, 5, 11, 4, 26, 13, 13, 13, 16, 24, 4, 13, 14, 23, 23, 14, 13, 20, 12, 20, 9, 12, 12, 9, 12, 1, 5, 28, 27, 26, 26, 26, 14, 7, 28, 7, 25, 9, 8, 17, 16, 7, 8, 23, 23, 7, 7, 3, 2, 9, 2, 9, 11, 9, 10, 9, 29, 4, 11, 6, 8, 20, 19, 18, 10, 3, 25, 4, 5, 13, 13, 12, 18, 3, 12, 13, 12, 3, 3, 13, 13, 22, 11, 21, 8, 20, 20, 10, 8, 6, 7, 5, 5, 26, 26, 25, 5, 25, 15, 25, 27, 24, 24, 15, 15, 7, 16, 14, 13, 7, 6, 22, 21, 2, 14, 2, 15, 3, 2, 9, 10, 7, 28, 9, 8, 8, 9, 10, 16, 6, 29, 6, 27, 18, 18, 20, 25, 11, 5, 25, 25, 12, 11, 7, 9, 11, 2, 11, 4, 13, 13, 12, 24, 2, 10, 6, 6, 20, 20, 20, 9, 5, 30, 5, 20, 5, 5, 26, 27, 25, 13, 13, 14, 24, 25, 5, 4, 24, 14, 23, 25, 15, 14, 13, 18, 1, 13, 21, 21, 12, 13, 12, 15, 1, 12, 1, 9, 28, 28, 27, 7, 8, 26, 26, 27, 8, 7, 29, 27, 9, 25, 10, 18, 17, 18, 16, 28, 5, 8, 24, 24, 7, 8, 7, 14, 10, 2, 9, 3, 10, 9, 12, 11, 4, 10, 9, 8, 5, 4, 11, 13, 20, 8, 20, 29, 18, 19, 11, 10, 4, 25, 4, 5, 13, 13, 13, 16, 3, 18, 4, 9, 13, 13, 12, 25, 23, 3, 13, 12, 22, 23, 12, 11, 20, 9, 20, 32, 11, 10, 9, 12, 3, 8, 5, 5, 27, 27, 26, 27, 26, 5, 26, 14, 25, 25, 27, 24, 7, 24, 15, 17, 8, 7, 16, 16, 7, 13, 8, 18, 23, 23, 21, 14, 7, 14, 3, 2, 3, 3, 2, 12, 8, 11, 7, 30, 9, 9, 8, 29, 6, 9, 11, 10, 6, 7, 29, 28, 19, 27, 18, 10, 20, 21, 25, 27, 3, 5, 25, 24, 12, 12, 12, 18, 11, 9, 12, 10, 12, 11, 4, 5, 3, 13, 12, 13, 2, 3, 10, 10, 21, 7, 20, 32, 20, 20, 10, 8, 25, 30, 6, 6, 5, 5, 5, 30, 26, 27, 25, 5, 13, 14, 15, 27, 24, 26, 5, 5, 24, 24, 15, 15, 15, 25, 15, 6, 14, 13, 18, 17, 6, 13, 22, 20, 12, 12, 13, 20, 2, 15, 1, 12, 2, 1, 9, 9, 8, 28, 28, 28, 9, 8, 26, 28, 8, 27, 9, 16, 29, 29, 28, 20, 6, 26, 10, 9, 18, 17, 19, 25, 5, 29, 5, 14, 24, 25, 24, 7, 11, 8, 7, 8, 10, 10, 2, 4, 10, 4, 10, 4, 13, 12, 12, 24, 4, 10, 9, 9, 6, 5, 4, 30,20, 13, 20, 8, 20, 20, 30, 27, 5, 20, 11, 10, 4, 4, 26, 27, 13, 5,13, 7, 13, 13, 16, 19, 24, 18, 4, 4, 13, 14, 14, 13, 23}
さて、例によって愚問を発する。
Q.もっとも素数の含有率の高いｎはどんなものであろうか？
A.１００００まででは｛２，３，７｝が1/2であり素数含有量が最大である。
予想：コラッツの系列での素数含有率の最大は1/2である。
ちなみに素数含有率とはコラッツの系列がｍ個であるときに素数がｎ個だったとすると、n/mである。
上記はどのような分布になるであろうか？
縦軸が素数含有率、横軸がｎである。この分布図が予想の可視化になっている。
1/3の辺りに対称軸があるような気配だ。
