A sumproduct number is a number such that the sum of 's digits times the product of 's digit is itself, for example
(1)

Obviously, such a number must be divisible by its digits as well as the sum of its digits. There are only three sumproduct numbers: 1, 135, and 144 (OEIS A038369). This can be demonstrated using the following argument due to D. Wilson.
Let be a digit sumproduct number, and let and be the sum and product of its digits. Because is a digit number, we have
(2)

Now, since is a sumproduct number, we have , giving
(3)

The inequality is fulfilled only by , so a sumproduct number has at most 84 digits.
This gives
(4)

Now, since is a product of digits, must be of the form . However, if 10 divides , then it also divides . This means that ends in 0 so the product of its digit is , giving . Hence we need not consider divisible by 10, and can assume is either of the form or . This reduces the search space for sumproduct numbers to a tractable size, and allowed Wilson to verify that there are no further sumproduct numbers.
The following table summarizes near misses up to , where is the sum and the product of decimal digits of .
OEIS  
0  A038369  1, 135, 144 
1  13, 91, 1529  
2  2, 32, 418, 3572, 32398, 66818, 1378946, ...  
3  219, 6177, 35277, 29859843, ...  
4  724, 1628, 5444, 437476, 1889285, 3628795, ...  
5  1285, 3187, 12875, 124987, 437467, 1889285, 3628795, ...  
6  3, 12, 14, 22, 42, 182, 1356, 1446, 7932, 18438, 25926, 29859834, ...  
7  23, 3463, 8633, 58247, 29719879, ...  
8  7789816, ...  
9  11, 81, 5871, 58329, ... 
The smallest values of whose sumproduct differs from by 0, 1, 2, ... are 1, 13, 2, 219, 724, 1285, 3, 23, 7789816, ... (OEIS A114457). The first unknown value occurs for , which must be greater than (E. W. Weisstein, Jan. 31, 2006).