Definition:Friedman Number

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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, $011 = 10 + 1$.


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

\(\displaystyle 25\) \(=\) \(\displaystyle 5^2\)
\(\displaystyle 121\) \(=\) \(\displaystyle 11^2\)
\(\displaystyle 125\) \(=\) \(\displaystyle 5^{\left({1 + 2}\right)}\)
\(\displaystyle 126\) \(=\) \(\displaystyle 21 \times 6\)
\(\displaystyle 127\) \(=\) \(\displaystyle 2^7 − 1\)
\(\displaystyle 128\) \(=\) \(\displaystyle 2^{\left({8 - 1}\right)}\)
\(\displaystyle 153\) \(=\) \(\displaystyle 51 \times 3\)

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.