A sum-product number is an integer that in a given base is equal to the sum of its digits times the product of its digits. Or, to put it algebraically, given an integer n that is l digits long in base b (with dx representing the xth digit), if
then n is a sum-product number in base b. In base 10, the only sum-product numbers are 0, 1, 135, 144 (sequence A038369 in the OEIS). Thus, for example, 144 is a sum-product number because 1 + 4 + 4 = 9, and 1 × 4 × 4 = 16, and 9 × 16 = 144.
1 is a sum-product number in any base, because of the multiplicative identity. 0 is also a sum-product number in any base, but no other integer with significant zeroes in the given base can be a sum-product number. 0 and 1 are also unique in being the only single-digit sum-product numbers in any given base; for any other single-digit number, the sum of the digits times the product of the digits works out to the number itself squared.
Any integer shown to be a sum-product number in a given base must, by definition, also be a Harshad number in that base.
In binary, 0 and 1 are the only sum-product numbers. The following table lists the sum-product numbers in bases up to 40 (using A−Z to represent digits 10 to 35):
The finiteness of the list for base 10 was proven by David Wilson. First he proved that a base 10 sum-product number will not have more than 84 digits. Next, he ruled out numbers with significant zeroes. Thereafter he concentrated on digit products of the forms
From Wilson's proof, Raymond Puzio developed the proof that in any positional base system there is only a finite set of sum-product numbers. First he observed that any number n of length l must satisfy
In Roman numerals, the only sum-product numbers are 1, 2, 3, and possibly 4 (if written IIII).