有趣的数

Python Day 3: Finding Fun Numbers

数学界有个关于 1729的故事, the Ramanujan Number.
一天 Hardy 去医院看 Ramanujan， 他说来的路上搭了一辆牌照很没有意思的士, 1729. Ramanujan 说: 不，这个数非常有意思; it is the smallest number expressible as the sum of two cubes in two different ways. （这是一个可以用两个立方和表达的最小数）

1729 = 1³ + 12³ = 9³ + 10³

我们今天用 python 来找一下有趣的数， the sum of divisors' square number（因子平方和完全平方数）, and perfect number （完美数）.

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GraphicMama-team / Pixabay[/caption]

Perfect Number: a perfect number is a positive integer that is equal to the sum of its proper positive divisors （除自己以外的因子和相加等于自己）

Example: the proper divisor of 6 is 1,2,3 and 6 = 1+2+3.

Python code:

#we want to find all the perfect number within 1000.
sieve= [1]*(1000+1)

n = 2
while n <= 1000:
    # check n
    if sieve[n] == n:
        print(n, "is a perfect number")
    # add n to all k * n where k > 1
    kn = 2 * n
    while kn <= 1000:
        sieve[kn] += n
        kn += n
    n += 1

Result:

6 is a perfect number
28 is a perfect number
496 is a perfect number

We can see 6,28 and 496 are perfect numbers.

[caption id="attachment_1415" align="alignnone" width="1024"] image

InspiredImages / Pixabay[/caption]

The sum of divisors' square number

Suppose we want to find some number such that the Sum of the squares of the number's divisors is a perfect square number

Example:

The divisors of 42 are 1,2,3,6,7,14,21,42

the square of divisors are 1,4,9,36,49, 196, 441, 1764

Sum of the square of divisors are 1+4+9+ 36+49+196+441+1764=2500

2500=500 * 500 is a perfect square

Python Code:

def list_squared(m,n):
    list=[]
    for i in range(m,n+1):
        sum=0
        s_list=[]
        for j in range(1,int(i**0.5)+1):
            if i%j==0:
                div=i/j
                sum+=j**2
                if j!=div:
                    sum+=div**2
        sqt=sum**0.5
        if int(sqt)==sqt:
            s_list=[i,sum]
            list.append(s_list)
    return list

list_squared(42,250)

Output:

[[42, 2500.0], [246, 84100.0]]

We can see 42, and 246 satisfy the conditions in the range (42,250)

我喜欢数论，因为它赋予了每一个数字特殊的意义 :) ， 最近两年，朋友和我都会对已经过去的和未来要来得年纪有一些敏感和伤感. 但我看这些数的时候，每一个都那么特别，都那么美:

27, the Cube age 立方年,
28, the Perfect age 完美年,
29, a Gaussian age 高斯年,
30, a smallest 2,3,5, divisor age 因子年,
31, a PRIME age 最好年 ,
32, a Binary age 二进制年!!!

I like my number theory classmate JOHNES said:

Appreciated every little things in life, like Tuesdays, it only comes once a week!
要感恩小事情哦， 像星期二，一周才一次 ：）

Happy Studying! 🐼