# Sum-product number

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 sum-product numbers (sequence A038369 in the OEIS): 0, 1, 135, and 144.

## Definition

Let $n$ be a natural number. We define the sum-product function for base $b>1$, $F_{b}:\mathbb {N} \rightarrow \mathbb {N}$, 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 $b$.
| 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 | $\varnothing$ | |
| 28 | 15B | |
| 29 | $\varnothing$ | |
| 30 | 976, 85MDA | |
| 31 | 44, 13H, 1E5 | |
| 32 | $\varnothing$ | |
| 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.