Collatz, Euler, Prime Stair

.py

#%%
def collatz():
    for i in range(1, 1000000):
        ns=[i]
        while(ns[-1]!=1):
            ns.append(3*ns[-1]+1 if ns[-1]%2==1 else ns[-1]/2)
        print(i, len(ns))

#%%
import sympy
def euler(n):
    if n==1:
        return 0
    ret = n
    for p in sympy.factorint(n).keys():
        ret *= 1-1/sympy.expand(p)
    return ret

euler(1050)

#%%
import sympy
import numpy as np
cnt = 0
data = []
for k in range(2, 100000):
    if sympy.isprime(k):
        cnt+= 1
    d= [k, cnt, k/np.log(k)]
    data.append(d)
    d.append((d[2]-d[1])/d[1])
import pandas as pd
df= pd.DataFrame(data)
import matplotlib.pyplot as plt
plt.grid()
plt.plot(df.loc[100:, 0], df.loc[100:, 1])
plt.plot(df.loc[100:, 0], df.loc[100:, 2])
ax=plt.twinx()
ax.plot(df.loc[100:,0], df.loc[100:,3])