# Definition:Friedman Number

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## Definition

A **Friedman number (base $n$)** is a (positive) integer which is the result of an expression in base $n$ arithmetic which contains exactly its digits.

The expression is subject to the following constraints:

- $(1): \quad$ The arithmetic operators $+$, $-$, $\times$, $\div$ and exponentiation are the only operators which are allowed.
- $(2): \quad$ Parentheses are allowed, but only in order to override the default operator precedence, otherwise every number would trivially be Friedman by $n = (n)$.
- $(3): \quad$ Leading zeroes are not allowed, otherwise other numbers would trivially be Friedman by, for example, $011 = 10 + 1$.

## Examples

The first few base $10$ Friedman numbers are as follows:

\(\displaystyle 25\) | \(=\) | \(\displaystyle 5^2\) | |||||||||||

\(\displaystyle 121\) | \(=\) | \(\displaystyle 11^2\) | |||||||||||

\(\displaystyle 125\) | \(=\) | \(\displaystyle 5^{\paren {1 + 2} }\) | |||||||||||

\(\displaystyle 126\) | \(=\) | \(\displaystyle 21 \times 6\) | |||||||||||

\(\displaystyle 127\) | \(=\) | \(\displaystyle 2^7 − 1\) | |||||||||||

\(\displaystyle 128\) | \(=\) | \(\displaystyle 2^{\paren {8 - 1} }\) | |||||||||||

\(\displaystyle 153\) | \(=\) | \(\displaystyle 51 \times 3\) |

This sequence is A036057 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Source of Name

This entry was named for Erich Friedman.