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{{Short description|Number equal to the product of the sum and product of its digits}}
{{Confuse|Product-over-sum}}
A '''sum-product number''' in a given [[radix|number base]] <math>b</math> is a [[natural number]] that is equal to the product of the [[digit sum|sum of its digits]] and the product of its digits. 

There are a finite number of sum-product numbers in any given base <math>b</math>. In [[base 10]], there are exactly four {{nowrap|sum-product}} numbers {{OEIS|id=A038369}}: 0, 1, 135, and 144.<ref>{{Cite OEIS|1=A038369|2=Numbers n such that n = (product of digits of n) * (sum of digits of n).}}</ref>

==Definition==
Let <math>n</math> be a natural number. We define the '''sum-product function''' for base <math>b > 1</math>, <math>F_b : \mathbb{N} \rightarrow \mathbb{N}</math>, to be the following:
: <math>F_b(n) = \left(\sum_{i = 1}^k d_i\right)\!\!\left(\prod_{j = 1}^k d_j\right)</math>
where <math>k = \lfloor \log_b{n} \rfloor + 1</math> is the number of digits in the number in base <math>b</math>, and 
: <math>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>n</math> is a '''{{nowrap|sum-product}} number''' if it is a [[Fixed point (mathematics)|fixed point]] for <math>F_b</math>, which occurs if <math>F_b(n) = n</math>. The natural numbers 0 and 1 are '''trivial {{nowrap|sum-product}} numbers''' for all <math>b</math>, and all other {{nowrap|sum-product}} numbers are '''nontrivial {{nowrap|sum-product}} numbers'''.

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

A natural number <math>n</math> is a '''sociable sum-product number''' if it is a [[periodic point]] for <math>F_b</math>, where <math>F_b^p(n) = n</math> for a positive [[integer]] <math>p</math>, and forms a [[Periodic sequence|cycle]] of period <math>p</math>. A {{nowrap|sum-product}} number is a sociable {{nowrap|sum-product}} number with <math>p = 1</math>, and an '''amicable {{nowrap|sum-product}} number''' is a sociable {{nowrap|sum-product}} number with <math>p = 2.</math>

All natural numbers <math>n</math> are [[periodic point#iterated function|preperiodic points]] for <math>F_b</math>, regardless of the base. This is because for any given digit count <math>k</math>, the minimum possible value of <math>n</math> is <math>b^{k-1}</math> and the maximum possible value of <math>n</math> is <math>b^k - 1 = \sum_{i=0}^{k-1} (b-1)^i.</math> The maximum possible digit sum is therefore <math>k(b-1)</math> and the maximum possible digit product is <math>(b-1)^k.</math> Thus, the {{nowrap|sum-product}} function value is <math>F_b(n) = k(b-1)^{k+1}.</math> This suggests that <math>k(b-1)^{k+1} \geq n \geq b^{k-1},</math> or dividing both sides by <math>(b-1)^{k-1}</math>, <math>k(b-1)^2 \geq {\left(\frac{b}{b-1}\right)}^{k-1}.</math> Since <math>\frac{b}{b-1} \geq 1,</math> this means that there will be a maximum value <math>k</math> where <math>{\left(\frac{b}{b-1}\right)}^k \leq k(b-1)^2,</math> because of the [[exponential function|exponential]] nature of <math>{\left(\frac{b}{b-1}\right)}^k</math> and the [[Linear function (calculus)|linearity]] of <math>k(b-1)^2.</math> Beyond this value <math>k</math>, <math>F_b(n) \leq n</math> always. Thus, there are a finite number of {{nowrap|sum-product}} numbers, and any natural number is guaranteed to reach a periodic point or a fixed point less than <math>b^k - 1,</math> making it a preperiodic point. 

The number of iterations <math>i</math> needed for <math>F_b^i(n)</math> to reach a fixed point is the {{nowrap|sum-product}} function's [[persistence of a number|persistence]] of <math>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 ''F''<sub>''b''</sub> for specific ''b''==
All numbers are represented in base <math>b</math>. 
{| class="wikitable"
! Base
! Nontrivial sum-product numbers
! Cycles
|-
| [[Base 2|2]] || (none) || (none)
|-
| [[Base 3|3]] || (none) || 2 → 11 → 2, 22 → 121 → 22
|-
| [[Base 4|4]] || 12 || (none)
|-
| [[Base 5|5]] || 341 || 22 → 31 → 22
|-
| [[Base 6|6]] || (none) || (none)
|-
| [[Base 7|7]] || 22, 242, 1254, 2343, 116655, 346236, 424644 ||
|-
| [[Base 8|8]] || (none) ||
|-
| [[Base 9|9]] || 13, 281876, 724856, 7487248 || 53 → 143 → 116 → 53
|-
| [[Base 10|10]] || 135, 144 ||
|-
| [[Base 11|11]] || 253, 419, 2189, 7634, 82974 ||
|-
| [[Base 12|12]] || 128, 173, 353 ||
|-
| [[Base 13|13]] || 435, A644, 268956 ||
|-
| [[Base 14|14]] || 328, 544, 818C ||
|-
| [[Base 15|15]] || 2585 ||
|-
| [[Base 16|16]] || 14 ||
|-
| [[Base 17|17]] || 33, 3B2, 3993, 3E1E, C34D, C8A2 ||
|-
| [[Base 18|18]] || 175, 2D2, 4B2 ||
|-
| [[Base 19|19]] || 873, B1E, 24A8, EAH1, 1A78A, 6EC4B7 ||
|-
| [[Base 20|20]] || 1D3, 14C9C, 22DCCG ||
|-
| [[Base 21|21]] || 1CC69 ||
|-
| [[Base 22|22]] || 24, 366C, 6L1E, 4796G ||
|-
| [[Base 23|23]] || 7D2, J92, 25EH6 ||
|-
| [[Base 24|24]] || 33DC ||
|-
| [[Base 25|25]] || 15, BD75, 1BBN8A ||
|-
| [[Base26|26]] || 81M, JN44, 2C88G, EH888 ||
|-
| [[Base 27|27]] || <math>\varnothing</math> ||
|-
| [[Base 28|28]] || 15B ||
|-
| [[Base 29|29]] || <math>\varnothing</math> ||
|-
| [[Base 30|30]] || 976, 85MDA ||
|-
| [[Base 31|31]] || 44, 13H, 1E5 ||
|-
| [[Base 32|32]] || <math>\varnothing</math> ||
|-
| [[Base 33|33]] || 1KS69, 54HSA ||
|-
| [[Base 34|34]] || 25Q8, 16L6W, B6CBQ ||
|-
| [[Base 35|35]] || 4U5W5 ||
|-
| [[Base 36|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 [[Cycle detection|to search for {{nowrap|sum-product}} numbers and cycles]] in [[Python (programming language)|Python]].
<syntaxhighlight lang="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
</syntaxhighlight>

==See also==
* [[Arithmetic dynamics#Other areas in which number theory and dynamics interact|Arithmetic dynamics]]
* [[Dudeney number]]
* [[Factorion]]
* [[Happy number]]
* [[Kaprekar's routine|Kaprekar's constant]]
* [[Kaprekar number]]
* [[Meertens number]]
* [[Narcissistic number]]
* [[Perfect digit-to-digit invariant]]
* [[Perfect digital invariant]]

==References==
{{reflist}}

{{Classes of natural numbers}}
[[Category:Arithmetic dynamics]]
[[Category:Articles with example Python (programming language) code]]
[[Category:Base-dependent integer sequences]]