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Amenable Number

A number $n$ which can be built up from Integers $a_1$, $a_2$, ..., $a_k$ by either Addition or Multiplication such that

\sum_{i=1}^k a_i=\prod_{i=1}^k a_i=n.

The numbers $\{a_1, \dots, a_n\}$ in the Sum are simply a Partition of $n$. The first few amenable numbers are

2+2=2\times 2&=&4\\
1+2+3=1\times 2\times 3&=&6\\
...times 4&=&8\\
1+1+2+2+2=1\times 1\times 2\times 2\times 2&=&8.

In fact, all Composite Numbers are amenable.

See also Composite Number, Partition, Sum


Tamvakis, H. ``Problem 10454.'' Amer. Math. Monthly 102, 463, 1995.

© 1996-9 Eric W. Weisstein