Perfect digit-to-digit invariant

Definition

Let n be a natural number which can be written in base b as the k-digit number d_{k-1}d_{k-2}...d_{1}d_{0} where each digit d_i is between 0 and b-1 inclusive, and n = \sum_{i=0}^{k-1} d_{i}b^{i}. We define the function F_b : \mathbb{N} \rightarrow \mathbb{N} as F_b(n) = \sum_{i=0}^{k - 1} {d_i}^{d_i}. (As 00 is usually undefined, there are typically two conventions used, one where it is taken to be equal to one, and another where it is taken to be equal to zero.{{cite book

F_b(1) = 1 for all b, and thus 1 is a trivial perfect digit-to-digit invariant in all bases, and all other perfect digit-to-digit invariants are nontrivial. For the second convention where 0^0 = 0, both 0 and 1 are trivial perfect digit-to-digit invariants.

A natural number n is a sociable digit-to-digit invariant if it is a periodic point for F_{b}, where F_{b}^k(n) = n for a positive integer k, and forms a cycle of period k. A perfect digit-to-digit invariant is a sociable digit-to-digit invariant with k = 1. An amicable digit-to-digit invariant is a sociable digit-to-digit invariant with k = 2.

All natural numbers n are preperiodic points for F_b, regardless of the base. This is because all natural numbers of base b with k digits satisfy b^{k-1} \leq n \leq (k){(b - 1)}^{b-1}. However, when k \geq b+1, then b^{k-1} (k){(b - 1)}^{b-1}, so any n will satisfy n F_b(n) until n . There are a finite number of natural numbers less than b^{b+1}, so the number is guaranteed to reach a periodic point or a fixed point less than b^{b+1}, making it a preperiodic point. This means also that there are a finite number of perfect digit-to-digit invariant and cycles for any given base b.

The number of iterations i needed for F_b^{i}(n) to reach a fixed point is the b-factorion function's persistence of n, and undefined if it never reaches a fixed point.

Perfect digit-to-digit invariants and cycles of F_b for specific b

All numbers are represented in base b.

Convention 0⁰ = 1

Convention 0⁰ = 0

Programming examples

The following program in Python determines whether an integer number is a Munchausen Number / Perfect Digit to Digit Invariant or not, following the convention 0^0 = 1.

num = int(input("Enter number:"))
temp = num
s = 0.0
while num > 0:
     digit = num % 10
     num //= 10
     s+= pow(digit, digit)
     
if s == temp:
    print("Munchausen Number")
else:
    print("Not Munchausen Number")

The examples below implement the perfect digit-to-digit invariant function described in the definition above to search for perfect digit-to-digit invariants and cycles in Python for the two conventions.

Convention 0⁰ = 1

def pddif(x: int, b: int) -> int:
    total = 0
    while x > 0:
        total = total + pow(x % b, x % b)
        x = x // b
    return total

def pddif_cycle(x: int, b: int) -> list[int]:
    seen = []
    while x not in seen:
        seen.append(x)
        x = pddif(x, b)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = pddif(x, b)
    return cycle

Convention 0⁰ = 0

def pddif(x: int, b: int) -> int:
    total = 0
    while x > 0:
        if x % b > 0:
            total = total + pow(x % b, x % b)
        x = x // b
    return total

def pddif_cycle(x: int, b: int) -> list[int]:
    seen = []
    while x not in seen:
        seen.append(x)
        x = pddif(x, b)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = pddif(x, b)
    return cycle

References

References

1. van Berkel, Daan. (2009). "On a curious property of 3435".
2. Olry, Regis and Duane E. Haines. [https://books.google.com/books?id=h30pAgAAQBAJ&pg=PA136 "Historical and Literary Roots of Münchhausen Syndromes"], from Literature, Neurology, and Neuroscience: Neurological and Psychiatric Disorders, Stanley Finger, Francois Boller, Anne Stiles, eds. Elsevier, 2013. p.136.
3. Daan van Berkel, [https://arxiv.org/abs/0911.3038 ''On a curious property of 3435.'']
4. (2014). "Things to Make and Do in the Fourth Dimension". Penguin UK.
5. [http://www.magic-squares.net/narciss.htm Narcisstic Number], Harvey Heinz