# sum-product number

A sum-product number is an integer $n$ that in a given base satisfies the equality

 $n=\sum_{i=1}^{m}d_{i}\prod_{i=1}^{m}d_{i}$

where $d_{i}$ is a digit of $n$, and $m$ is the number of digits of $n$. This means a test of whether the sum of the digits of $n$ times the product of the digits of $n$ is equal to $n$.

For example, the number 128 in base 7 is a sum-product number since

 $242_{7}=(2+4+2)(2\cdot 4\cdot 2)$

All sum-product numbers are Harshad numbers, too.

0 and 1 are sum-product numbers in any positional base. The proof that the set of sum-product numbers in base 2 is finite is elementary enough not to inspire claims of authorship. The proof that the set of sum-product numbers in base 10 is finite (specifically, 0, 1, 135 and 144) is more involved but within the realm of basic algebra, and it points the way to a formulation of the proof that number of sum-product numbers in any base is finite.

Title sum-product number SumproductNumber 2013-03-22 15:46:50 2013-03-22 15:46:50 Mravinci (12996) Mravinci (12996) 8 Mravinci (12996) Definition msc 11A63 sum product number