A sum-product number is a positive integer 
such that the sum
of its decimal digits times the product
of its decimal digits equals 
, 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 sum-product numbers: 1, 135, and 144. The sequence OEIS A038369 also includes 0, which satisfies the digit equation but is excluded by the positive-integer convention here. The completeness of the positive list can be demonstrated using the following argument due to D. Wilson.
Let 
be a 
-digit sum-product number, and let 
and 
be the sum and product
of its digits. Because 
is a 
-digit number, we have
 |
(2)
|
Now, since 
is a sum-product number, we have 
, giving
 |
(3)
|
The inequality 
is fulfilled only by 
, so a sum-product 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
digits 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 sum-product numbers
to a tractable size, and allowed Wilson to verify that there are no further sum-product
numbers.
The following table summarizes near misses among positive integers 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, ... | |
| 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 positive values of 
whose sum-product differs from 
by 0, 1, 2, ... are 1, 13, 2, 219, 724, 1285, 3, 23, 7789816,
... (OEIS A114457). The sequence
uses 0 when no such positive integer exists and
reports that no solution exists for difference 33 (OEIS A114457).
