Yuqing Ji

220 and 284

An amicable pair

220 and 284 is called amicable because the sum of the proper divisors of one is equal to the other.

220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

284: 1, 2, 4, 71, 142, 284

1 + 2 + 4 + 71 + 142 = 220

Why not include the negative number

For fun and there’s no reason we can’t do this. if including the negative number, will find:

The sum of the divisors of each negative number is its opposite

284: 1, -1, 2, -2, 4, -4, 71, -71, 142, -142, 284, -284

1 + (-1) + 2 + (-2) + 4 + (-4) + 71 + (-71) + 142 + (-142) + 284 = 284

Isn’t it perfect?

Some numbers can be with 2 friends

12, with -6 and -11

…a new world to explore

A method for factorization

get the prime factors first
then, combine the prime factors
then, get the factors which is divisible to the input

#!/usr/bin/env python

import pandas as pd
import itertools

class Factorization:

    def __init__(self, input, plist):
        self.input = input
        self.input_cp = input
        self.plist = plist

    def prime_facterise(self):
        pflist = []
        used_plist = []
        dupli_plist = []

        if self.input <= self.plist[-1]:
            for p in self.plist:
                while self.input % p == 0:
                    if p not in used_plist:
                        used_plist.append(p)
                    else:
                        dupli_plist.append(p)
                    self.input = self.input / p
            for i in used_plist:
                if i not in pflist:
                    pflist.append(i)
            allpf = pflist + dupli_plist

        if self.input_cp < 0:
            allpf += [(-1) * i for i in allpf]
            allpf.append(-1)
            pflist += [(-1) * i for i in pflist]
            pflist.append(-1)

        allpf.append(1)
        pflist.append(1)

        return allpf, pflist, dupli_plist

    def facterise(self):
        allpf = self.prime_facterise()[0]
        allf = []
        uflist = []

        for i in range(1, len(allpf) + 1):
            iter = itertools.combinations(allpf, i)
            allf.append(list(iter))

        f = 1
        flist = []
        if allf != []:
            for l in allf:
                for t in l:
                    for i in t:
                        f *= i
                        flist.append(f)
                    f = 1

        for f in flist:
            if self.input_cp % f == 0:
                uflist.append(f)

        uflist = list(set(uflist))

        return uflist

df = pd.read_csv('./prime1000.csv')
plist = df['prime'].tolist()

# example
f = Factorization(220, plist)
factors = f.factorize()

About | Blog | Email | Github | 中文