Sum-product number

Definition

Let n be a natural number. We define the sum-product function for base b 1, F_b : \mathbb{N} \rightarrow \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{n \bmod{b^{i+1}} - n \bmod b^i}{b^i} is the value of each digit of the number. A natural number n is a sum-product 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 sum-product numbers for all b, and all other sum-product numbers are nontrivial sum-product 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 sum-product number is a sociable sum-product number with p = 1, and an amicable sum-product number is a sociable sum-product 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 sum-product function value is F_b(n) = k(b-1)^{k+1}. This suggests that k(b-1)^{k+1} \geq n \geq b^{k-1}, or dividing both sides by (b-1)^{k-1}, k(b-1)^2 \geq {\left(\frac{b}{b-1}\right)}^{k-1}. Since \frac{b}{b-1} \geq 1, this means that there will be a maximum value k where {\left(\frac{b}{b-1}\right)}^k \leq 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) \leq n always. Thus, there are a finite number of sum-product 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 sum-product 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.

Sum-product numbers and cycles of ''F''_''b'' for specific ''b''

All numbers are represented in base b.

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 to search for sum-product 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

References

References

1. {{Cite OEIS