Probably an interesting question is why it seems that with base $2$ instead of $10$ and the same procedure, we get large primes: (Notice that for $d=2$ the starting number $20$ gets a prime).

```
def ff(n,d=10):
ll = []
for p in sorted(prime_divisors(n)):
dd = Integer(p).digits(d)
dd.reverse()
ll.extend(dd)
if valuation(n,p)>1:
dd = Integer(valuation(n,p)).digits(d)
dd.reverse()
ll.extend(dd)
return sum(ll[len(ll)-1-i]*d**i for i in range(len(ll)))
def iter_ff(n,k,d):
if k==0:
return n
else:
return ff(iter_ff(n,k-1,d),d)
x = 2**2*5
while not is_prime(Integer(x)):
x = ff(x,d=2)
print(x)
maxIter = 55
for d in range(2,11):
for n in range(2,100):
x = n
i = 0
while not is_prime(Integer(x)) and i < maxIter:
x = ff(x,d=d)
print("base = ",d,"starting number = ",n,"current number = ",x,"iteration count = ", i+1)
i+=1
#print(d,n,iter_ff(n,k,d),is_prime(Integer(iter_ff(n,k,d))))
```

Output:

```
85
177
251
base = 2 starting number = 4 current number = 10 iteration count = 1
base = 2 starting number = 4 current number = 21 iteration count = 2
base = 2 starting number = 4 current number = 31 iteration count = 3
base = 2 starting number = 6 current number = 11 iteration count = 1
base = 2 starting number = 8 current number = 11 iteration count = 1
base = 2 starting number = 9 current number = 14 iteration count = 1
base = 2 starting number = 9 current number = 23 iteration count = 2
base = 2 starting number = 10 current number = 21 iteration count = 1
base = 2 starting number = 10 current number = 31 iteration count = 2
base = 2 starting number = 12 current number = 43 iteration count = 1
base = 2 starting number = 14 current number = 23 iteration count = 1
base = 2 starting number = 15 current number = 29 iteration count = 1
base = 2 starting number = 16 current number = 20 iteration count = 1
base = 2 starting number = 16 current number = 85 iteration count = 2
base = 2 starting number = 16 current number = 177 iteration count = 3
base = 2 starting number = 16 current number = 251 iteration count = 4
base = 2 starting number = 18 current number = 46 iteration count = 1
base = 2 starting number = 18 current number = 87 iteration count = 2
base = 2 starting number = 18 current number = 125 iteration count = 3
base = 2 starting number = 18 current number = 23 iteration count = 4
base = 2 starting number = 20 current number = 85 iteration count = 1
base = 2 starting number = 20 current number = 177 iteration count = 2
base = 2 starting number = 20 current number = 251 iteration count = 3
base = 2 starting number = 21 current number = 31 iteration count = 1
base = 2 starting number = 22 current number = 43 iteration count = 1
base = 2 starting number = 24 current number = 47 iteration count = 1
base = 2 starting number = 25 current number = 22 iteration count = 1
base = 2 starting number = 25 current number = 43 iteration count = 2
base = 2 starting number = 26 current number = 45 iteration count = 1
base = 2 starting number = 26 current number = 117 iteration count = 2
base = 2 starting number = 26 current number = 237 iteration count = 3
base = 2 starting number = 26 current number = 463 iteration count = 4
base = 2 starting number = 27 current number = 15 iteration count = 1
base = 2 starting number = 27 current number = 29 iteration count = 2
base = 2 starting number = 28 current number = 87 iteration count = 1
base = 2 starting number = 28 current number = 125 iteration count = 2
base = 2 starting number = 28 current number = 23 iteration count = 3
base = 2 starting number = 30 current number = 93 iteration count = 1
base = 2 starting number = 30 current number = 127 iteration count = 2
base = 2 starting number = 32 current number = 21 iteration count = 1
base = 2 starting number = 32 current number = 31 iteration count = 2
base = 2 starting number = 33 current number = 59 iteration count = 1
base = 2 starting number = 34 current number = 81 iteration count = 1
base = 2 starting number = 34 current number = 28 iteration count = 2
base = 2 starting number = 34 current number = 87 iteration count = 3
base = 2 starting number = 34 current number = 125 iteration count = 4
base = 2 starting number = 34 current number = 23 iteration count = 5
base = 2 starting number = 35 current number = 47 iteration count = 1
base = 2 starting number = 36 current number = 174 iteration count = 1
base = 2 starting number = 36 current number = 381 iteration count = 2
base = 2 starting number = 36 current number = 511 iteration count = 3
```

For $n=217$ and for $d=2$ you get as an example a cycle which is not a prime:

SageMath-Script

$$217, 255, 945, 1007, 1269,1007,1269,1007\ldots$$

10more comments