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Sum-Product Number


A sum-product number is a number n such that the sum of n's digits times the product of n's digit is n itself, for example

 135=(1+3+5)(1·3·5).
(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 n be a d-digit sum-product number, and let s and p be the sum and product of its digits. Because n is a d-digit number, we have

 10^(d-1)<=n;  s<=9d;p<=9^d.
(2)

Now, since n is a sum-product number, we have n=sp, giving

 10^(d-1)<=n=sp<=(9d)(9^d).
(3)

The inequality 10^(d-1)<=(9d)(9^d) is fulfilled only by d<=84, so a sum-product number has at most 84 digits.

This gives

 s<=9d<=756;  p<=n<10^(85).
(4)

Now, since p is a product of digits, p must be of the form 2^a3^b5^c7^d. However, if 10 divides p, then it also divides n. This means that n ends in 0 so the product of its digit is p=0, giving n=sp=0. Hence we need not consider p divisible by 10, and can assume p is either of the form 2^a3^b7^c or 3^a5^b7^c. 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 10^8, where S(n) is the sum and P(n) the product of decimal digits of n.

|S(n)P(n)-n|OEISn
0A0383691, 135, 144
113, 91, 1529
22, 32, 418, 3572, 32398, 66818, 1378946, ...
3219, 6177, 35277, 29859843, ...
4724, 1628, 5444, 437476, 1889285, 3628795, ...
51285, 3187, 12875, 124987, 437467, 1889285, 3628795, ...
63, 12, 14, 22, 42, 182, 1356, 1446, 7932, 18438, 25926, 29859834, ...
723, 3463, 8633, 58247, 29719879, ...
87789816, ...
911, 81, 5871, 58329, ...

The smallest values of n whose sum-product differs from n by 0, 1, 2, ... are 1, 13, 2, 219, 724, 1285, 3, 23, 7789816, ... (OEIS A114457). The first unknown value occurs for n=33, which must be greater than 9.4×10^(10) (E. W. Weisstein, Jan. 31, 2006).


See also

Amenable Number, Digit, Harshad Number

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References

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

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Sum-Product Number

Cite this as:

Weisstein, Eric W. "Sum-Product Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Sum-ProductNumber.html

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