Sum-product number

Template:Short description Template:Confuse A sum-product number in a given number base $b$ is a natural number that is equal to the product of the sum of its digits and the product of its digits.

There are a finite number of sum-product numbers in any given base $b$. In base 10, there are exactly four Template:Nowrap numbers Template:OEIS: 0, 1, 135, and 144.

Definition

Let $n$ be a natural number. We define the sum-product function for base $b>1$, $F_b:\mathbb{N}\to \mathbb{N}$, to be the following:

$F_b(n)=\left(\sum _{i=1}^k{d}_i\right)\left(\prod _{j=1}^k{d}_j\right)$

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

$d_i=\frac{nmod{b}^{i+1}-n{modb}^i}{b^i}$

is the value of each digit of the number. A natural number $n$ is a Template:Nowrap number if it is a fixed point for $F_b$, which occurs if $F_b(n)=n$. The natural numbers 0 and 1 are trivial Template:Nowrap numbers for all $b$, and all other Template:Nowrap numbers are nontrivial Template:Nowrap numbers.

For example, the number 144 in base 10 is a sum-product number, because $1+4+4=9$, $1\times 4\times 4=16$, and $9\times 16=144$.

A natural number $n$ is a sociable sum-product number if it is a periodic point for $F_b$, where $F_b^p(n)=n$ for a positive integer $p$, and forms a cycle of period $p$. A Template:Nowrap number is a sociable Template:Nowrap number with $p=1$, and an amicable Template:Nowrap number is a sociable Template:Nowrap number with $p=2.$

All natural numbers $n$ are preperiodic points for $F_b$, regardless of the base. This is because for any given digit count $k$, the minimum possible value of $n$ is $b^{k-1}$ and the maximum possible value of $n$ is $b^k-1=\sum _{i=0}^{k-1}(b-1{)}^i.$ The maximum possible digit sum is therefore $k(b-1)$ and the maximum possible digit product is $(b-1{)}^k.$ Thus, the Template:Nowrap function value is $F_b(n)=k(b-1{)}^{k+1}.$ This suggests that $k(b-1{)}^{k+1}\ge n\ge b^{k-1},$ or dividing both sides by $(b-1{)}^{k-1}$, $k(b-1{)}^2\ge {\left(\frac{b}{b-1}\right)}^{k-1}.$ Since $\frac{b}{b-1}\ge 1,$ this means that there will be a maximum value $k$ where ${\left(\frac{b}{b-1}\right)}^k\le k(b-1{)}^2,$ because of the exponential nature of ${\left(\frac{b}{b-1}\right)}^k$ and the linearity of $k(b-1{)}^2.$ Beyond this value $k$, $F_b(n)\le n$ always. Thus, there are a finite number of Template:Nowrap numbers, and any natural number is guaranteed to reach a periodic point or a fixed point less than $b^k-1,$ making it a preperiodic point.

The number of iterations $i$ needed for $F_b^i(n)$ to reach a fixed point is the Template:Nowrap function's persistence of $n$, and undefined if it never reaches a fixed point.

Any integer shown to be a sum-product number in a given base must, by definition, also be a Harshad number in that base.

Extension to negative integers

Sum-product numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.

Programming example

The example below implements the sum-product function described in the definition above [[Cycle detection|to search for Template:Nowrap numbers and cycles]] in Python.

def sum_product(x: int, b: int) -> int:
    """Sum-product number."""
    sum_x = 0
    product = 1
    while x > 0:
        if x % b > 0:
            sum_x = sum_x + x % b
            product = product * (x % b)
        x = x // b
    return sum_x * product

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

See also

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