Sum-Product Number

A sum-product 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 sum-product numbers: 1, 135, and 144 (OEIS A038369). This 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 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 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 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 sum-product 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).

REFERENCES:

Sloane, N. J. A. Sequences A038369 and A114457 in "The On-Line Encyclopedia of Integer Sequences."