Perfect digit-to-digit invariant

Let ${\displaystyle n}$ be a natural number. We define the perfect digit-to-digit invariant function for base ${\displaystyle b>1}$ ${\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }$ to be the following:

${\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}{d_{i}}^{d_{i}}}$.

where ${\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}$ is the number of digits in the number in base ${\displaystyle b}$ and

${\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}}$

is the value of each digit of the number. 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.[5][6] A natural number ${\displaystyle n}$ is a perfect digit-to-digit invariant if it is a fixed point for ${\displaystyle F_{b}}$, which occurs if ${\displaystyle F_{b}(n)=n}$. For the first convention, ${\displaystyle 1}$ is a fixed point for all ${\displaystyle b}$, and thus is a trivial perfect digit-to-digit invariant for all ${\displaystyle b}$, and all other perfect digit-to-digit invariants are nontrivial perfect digit-to-digit invariants. For the second convention, both ${\displaystyle 0}$ and ${\displaystyle 1}$ are trivial perfect digit-to-digit invariants.

For example, the number 3435 in base ${\displaystyle b=10}$ is a perfect digit-to-digit invariant because ${\displaystyle 3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435}$.

For ${\displaystyle b=2}$, in the first convention ${\displaystyle 0^{0}=1}$, ${\displaystyle F_{b}(n)}$ is simply the number of digits ${\displaystyle k}$ in the base 2 representation, and in the second convention ${\displaystyle 0^{0}=0}$, ${\displaystyle F_{b}(n)}$ is simply the digit sum.

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

All natural numbers ${\displaystyle n}$ are preperiodic points for ${\displaystyle F_{b}}$, regardless of the base. This is because all natural numbers of base ${\displaystyle b}$ with ${\displaystyle k}$ digits satisfy ${\displaystyle b^{k-1}\leq n\leq (k){(b-1)}^{b-1}}$. However, when ${\displaystyle k\geq b+1}$, then ${\displaystyle b^{k-1}>(k){(b-1)}^{b-1}}$, so any ${\displaystyle n}$ will satisfy ${\displaystyle n>F_{b}(n)}$ until ${\displaystyle n<b^{b+1}}$. There are a finite number of natural numbers less than ${\displaystyle b^{b+1}}$, so the number is guaranteed to reach a periodic point or a fixed point less than ${\displaystyle b^{b+1}}$, 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 ${\displaystyle b}$.

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

All numbers are represented in base ${\displaystyle b}$.

Convention ${\displaystyle 0^{0}=1}$

Base	Nontrivial perfect digit-to-digit invariants (${\displaystyle n\neq 1}$)	Cycles
2	10	${\displaystyle \varnothing }$
3	12, 22	2 → 11 → 2
4	131, 313	2 → 10 → 2
5	${\displaystyle \varnothing }$	2 → 4 → 2011 → 12 → 10 → 2 104 → 2013 → 113 → 104
6	22352, 23452	4 → 1104 → 1111 → 4 23445 → 24552 → 50054 → 50044 → 24503 → 23445
7	13454	12066 → 536031 → 265204 → 265623 → 551155 → 51310 → 12125 → 12066
8		405 → 6466 → 421700 → 3110776 → 6354114 → 142222 → 421 → 405
9	31, 156262, 1656547
10	3435
11
12	3A67A54832

Convention ${\displaystyle 0^{0}=0}$

Base	Nontrivial perfect digit-to-digit invariants (${\displaystyle n\neq 0}$, ${\displaystyle n\neq 1}$)[1]	Cycles
2	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
3	12, 22	2 → 11 → 2
4	130, 131, 313	${\displaystyle \varnothing }$
5	103, 2024	2 → 4 → 2011 → 11 → 2 9 → 2012 → 9
6	22352, 23452	5 → 22245 → 23413 → 1243 → 1200 → 5 53 → 22332 → 150 → 22250 → 22305 → 22344 → 2311 → 53
7	13454
8	400, 401
9	30, 31, 156262, 1647063, 1656547, 34664084
10	3435, 438579088
11	${\displaystyle \varnothing }$	${\displaystyle \varnothing }$
12	3A67A54832

The examples below implements 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.

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

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

• Arithmetic dynamics
• Dudeney number
• Factorion
• Happy number
• Kaprekar's constant
• Kaprekar number
• Meertens number
• Narcissistic number
• Perfect digital invariant
• Sum-product number

• Parker, Matt. "3435". Numberphile. Brady Haran. Archived from the original on 2017-04-13. Retrieved 2013-04-01.