Sum-product number

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 d_x representing the xth digit), if

$n=(\sum _{i=1}^{l}d_{i})(\prod _{j=1}^{l}d_{j})$

then n is a sum-product number in base b. In base 10, the only sum-product numbers are 0, 1, 135, 144 (sequence 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):

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Wikipedia