Starting with a positive integer $x$, find the sum of the $n^{th}$ powers of the prime factors of $x$, including multiplicities. Then find the sum of the $n^{th}$ prime factors of the result etc. until an $n^{th}$ power of a prime is reached. Will the sequence always terminate, or can it get caught in a loop or diverge to infinity?
python code for creating this sequence:
def factor(n): m,p,r,k=n,3,7,[] while m%2==0: k.append(2) r=1 m=m/2 while p<=n**0.5 and m!=1: if m%p==0: k.append(p) r=1 m=m/p else: p+=2 if r==1: while p<=n and m!=1: if m%p==0: k.append(p) m=m/p else: p+=2 return k y=1 x,n=int(input("start")),int(input("power")) while x!=y and x!=0: y,x=x,0 t=factor(y) for e in t: x+=e**n print(x)