考拉兹猜想

考拉兹猜想是1927年提出的猜想，至今仍然未解。考拉兹猜想的内容是：对于每一个正整数，如果他是奇数，则对它乘3再加1，如果它是偶数，则对他除2,。这样循环，最终结果都能得到1

第一次见到这个猜想是在《The Pratice of Computing Using Python》（中文版《Python入门经典-已解决计算问题为导向的Python编程实践》）。对于这个猜想，作者设计了如下程序（验证一个数是否符合猜想）：

# Generating a Hailstone sequence
numString = raw_input("Enter a positive integrate:")
num = int(numString)
count = 0

print "Starting wiht number:", num
print "Sequence is: ",

while num > 1: # stop when the sequence reaches 1
if num%2: # num is odd
num = num*3 + 1
else: # num is even
num = num/2
print num, ",", # add num to sequence

count += 1 # add num to count

else:
print # blank line fo nicer output
print "Sequence is ", count, " numbers long"

为了能够用图直观的表示出数列的收敛趋势，我对程序进行了如下的修改：

# -*- coding:utf-8 -*-
# Filename: collatz_guess.py

# 获得输入
numString = raw_input("输入一个数：")
num = int(numString)

# 建立一个计数列表和结果计算过程列表
count = 0
count_list = [0]
sequence_list = []
sequence_list.append(num)

while num != 1:
if num%2:
num = num*3 + 1 # 如果是奇数，乘以3，再加1
else:
num = num/2 # 如果是偶数，除以2
count += 1
count_list.append(count) # 把计数添加到计数序列
sequence_list.append(num) # 把结果添加到计算过程序列
else:
print
print sequence_list

# 绘图
import numpy as np
import pylab as pl
from scipy import interpolate

x = np.linspace(0, count, count*10+1) # 根据count设置差值变量的范围
count_array = np.array(count_list)
sequence_array = np.array(sequence_list) # 把数组转换为array
pl.plot(count_array, sequence_array, 'ro') # 绘制count， num的散点图
f = interpolate.interp1d(count_array, sequence_array, kind="slinear")
y = f(x)
pl.plot(x, y, label = "collatz guess") # 插值并绘图

pl.legend(loc='hight right')
pl.show()

修改后的程序，需要加载numpy，和scipy。