A sum-product number in a given number base 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 . In base 10, there are exactly four numbers : 0, 1, 135, and 144.
Let be a natural number. We define the sum-product function for base , , to be the following:
where is the number of digits in the number in base , and
is the value of each digit of the number. A natural number is a number if it is a fixed point for , which occurs if . The natural numbers 0 and 1 are trivial numbers for all , and all other numbers are nontrivial numbers.
For example, the number 144 in base 10 is a sum-product number, because , , and .
A natural number is a sociable sum-product number if it is a periodic point for , where for a positive integer , and forms a cycle of period . A number is a sociable number with , and an amicable number is a sociable number with
All natural numbers are preperiodic points for , regardless of the base. This is because for any given digit count , the minimum possible value of is and the maximum possible value of is The maximum possible digit sum is therefore and the maximum possible digit product is Thus, the function value is This suggests that or dividing both sides by , Since this means that there will be a maximum value where because of the exponential nature of and the linearity of Beyond this value , always. Thus, there are a finite number of numbers, and any natural number is guaranteed to reach a periodic point or a fixed point less than making it a preperiodic point.
The number of iterations needed for to reach a fixed point is the function's persistence of , 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.
All numbers are represented in base .
Sum-product numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
The example below implements the sum-product function described in the definition above to search for numbers and cycles in Python.