# 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 |