sum-product number

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


where di 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


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