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# Friedman numbers

Please do not rely on any information it contains.

A Friedman number in a given base ${\displaystyle b}$ is a number that can be written in a nontrivial way using their digits with the basic operations of arithmetic.

Here are a few examples in base 10:

• ${\displaystyle 25=5^{2}}$
• ${\displaystyle 121=11^{2}}$
• ${\displaystyle 125=5^{2+1}}$
• ${\displaystyle 126=6\times 21}$
• ${\displaystyle 127=2^{7}-1}$
• ${\displaystyle 128=2^{8-1}}$
• ${\displaystyle 153=3\times 51}$
• ${\displaystyle 216=6^{2+1}}$
• ${\displaystyle 289=(8+9)^{2}}$

See A036057 for more base 10 Friedman numbers.

A Friedman number is said to be "orderly" when it is possible to put the digits in the expression in the same order as in the usual representation. For example, with ${\displaystyle 2^{7}-1}$, we can rearrange like so: ${\displaystyle -1+2^{7}}$, hence 127 is orderly. See A080035 for more base 10 orderly Friedman numbers.

The concept can also be extended to Roman numerals. For example, LXXXI = IXX × X/L. In many cases, inconvenient "digits" can be "neutralized" by making them into exponents for I, such as in XLIX = L – IXX. See A195419 for more Roman Friedman numbers.