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
Canonical name	SumproductNumber
Date of creation	2013-03-22 15:46:50
Last modified on	2013-03-22 15:46:50
Owner	Mravinci (12996)
Last modified by	Mravinci (12996)
Numerical id	8
Author	Mravinci (12996)
Entry type	Definition
Classification	msc 11A63
Synonym	sum product number