Sum-product number
A sum-product number in a given number base [math]\displaystyle{ b }[/math] 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 [math]\displaystyle{ b }[/math]. In base 10, there are exactly four sum-product numbers (sequence A038369 in the OEIS): 0, 1, 135, and 144.[1]
Definition
Let [math]\displaystyle{ n }[/math] be a natural number. We define the sum-product function for base [math]\displaystyle{ b \gt 1 }[/math], [math]\displaystyle{ F_b : \mathbb{N} \rightarrow \mathbb{N} }[/math], to be the following:
- [math]\displaystyle{ F_b(n) = \left(\sum_{i = 1}^k d_i\right)\!\!\left(\prod_{j = 1}^k d_j\right) }[/math]
where [math]\displaystyle{ k = \lfloor \log_b{n} \rfloor + 1 }[/math] is the number of digits in the number in base [math]\displaystyle{ b }[/math], and
- [math]\displaystyle{ d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i} }[/math]
is the value of each digit of the number. A natural number [math]\displaystyle{ n }[/math] is a sum-product number if it is a fixed point for [math]\displaystyle{ F_b }[/math], which occurs if [math]\displaystyle{ F_b(n) = n }[/math]. The natural numbers 0 and 1 are trivial sum-product numbers for all [math]\displaystyle{ b }[/math], 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 [math]\displaystyle{ 1 + 4 + 4 = 9 }[/math], [math]\displaystyle{ 1 \times 4 \times 4 = 16 }[/math], and [math]\displaystyle{ 9 \times 16 = 144 }[/math].
A natural number [math]\displaystyle{ n }[/math] is a sociable sum-product number if it is a periodic point for [math]\displaystyle{ F_b }[/math], where [math]\displaystyle{ F_b^p(n) = n }[/math] for a positive integer [math]\displaystyle{ p }[/math], and forms a cycle of period [math]\displaystyle{ p }[/math]. A sum-product number is a sociable sum-product number with [math]\displaystyle{ p = 1 }[/math], and an amicable sum-product number is a sociable sum-product number with [math]\displaystyle{ p = 2. }[/math]
All natural numbers [math]\displaystyle{ n }[/math] are preperiodic points for [math]\displaystyle{ F_b }[/math], regardless of the base. This is because for any given digit count [math]\displaystyle{ k }[/math], the minimum possible value of [math]\displaystyle{ n }[/math] is [math]\displaystyle{ b^{k-1} }[/math] and the maximum possible value of [math]\displaystyle{ n }[/math] is [math]\displaystyle{ b^k - 1 = \sum_{i=0}^{k-1} (b-1)^k. }[/math] The maximum possible digit sum is therefore [math]\displaystyle{ k(b-1) }[/math] and the maximum possible digit product is [math]\displaystyle{ (b-1)^k. }[/math] Thus, the sum-product function value is [math]\displaystyle{ F_b(n) = k(b-1)^{k+1}. }[/math] This suggests that [math]\displaystyle{ k(b-1)^{k+1} \geq n \geq b^{k-1}, }[/math] or dividing both sides by [math]\displaystyle{ (b-1)^{k-1} }[/math], [math]\displaystyle{ k(b-1)^2 \geq {\left(\frac{b}{b-1}\right)}^{k-1}. }[/math] Since [math]\displaystyle{ \frac{b}{b-1} \geq 1, }[/math] this means that there will be a maximum value [math]\displaystyle{ k }[/math] where [math]\displaystyle{ {\left(\frac{b}{b-1}\right)}^k \leq k(b-1)^2, }[/math] because of the exponential nature of [math]\displaystyle{ {\left(\frac{b}{b-1}\right)}^k }[/math] and the linearity of [math]\displaystyle{ k(b-1)^2. }[/math] Beyond this value [math]\displaystyle{ k }[/math], [math]\displaystyle{ F_b(n) \leq n }[/math] 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 [math]\displaystyle{ b^k - 1, }[/math] making it a preperiodic point.
The number of iterations [math]\displaystyle{ i }[/math] needed for [math]\displaystyle{ F_b^i(n) }[/math] to reach a fixed point is the sum-product function's persistence of [math]\displaystyle{ n }[/math], 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 Fb for specific b
All numbers are represented in base [math]\displaystyle{ b }[/math].
Base | Nontrivial sum-product numbers | Cycles |
---|---|---|
2 | (none) | (none) |
3 | (none) | 2 → 11 → 2, 22 → 121 → 22 |
4 | 12 | (none) |
5 | 341 | 22 → 31 → 22 |
6 | (none) | (none) |
7 | 22, 242, 1254, 2343, 116655, 346236, 424644 | |
8 | (none) | |
9 | 13, 281876, 724856, 7487248 | 53 → 143 → 116 → 53 |
10 | 135, 144 | |
11 | 253, 419, 2189, 7634, 82974 | |
12 | 128, 173, 353 | |
13 | 435, A644, 268956 | |
14 | 328, 544, 818C | |
15 | 2585 | |
16 | 14 | |
17 | 33, 3B2, 3993, 3E1E, C34D, C8A2 | |
18 | 175, 2D2, 4B2 | |
19 | 873, B1E, 24A8, EAH1, 1A78A, 6EC4B7 | |
20 | 1D3, 14C9C, 22DCCG | |
21 | 1CC69 | |
22 | 24, 366C, 6L1E, 4796G | |
23 | 7D2, J92, 25EH6 | |
24 | 33DC | |
25 | 15, BD75, 1BBN8A | |
26 | 81M, JN44, 2C88G, EH888 | |
27 | [math]\displaystyle{ \varnothing }[/math] | |
28 | 15B | |
29 | [math]\displaystyle{ \varnothing }[/math] | |
30 | 976, 85MDA | |
31 | 44, 13H, 1E5 | |
32 | [math]\displaystyle{ \varnothing }[/math] | |
33 | 1KS69, 54HSA | |
34 | 25Q8, 16L6W, B6CBQ | |
35 | 4U5W5 | |
36 | 16, 22O |
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
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
References
![]() | Original source: https://en.wikipedia.org/wiki/Sum-product number.
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