Munchausen number

A natural number is called a Münchhausen number if the sum of its individual digits raised to the power of themselves results in this number. For example, if the natural number has a decimal representation , the condition must be met for a Münchhausen number . ${\displaystyle n}$ ${\displaystyle n=a_{k}a_{k-1}\dots a_{0}}$ ${\displaystyle \textstyle n=\sum _{i=0}^{k}a_{i}^{a_{i}}}$

One example is . To calculate the powers of the digits, Munchausen numbers are usually defined in connection with . Then there are four Münchhausen numbers: ${\displaystyle 3435=3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125}$ ${\displaystyle 0^{0}=0}$

• 0,
• 1,
• 3435 and
• 438.579.088.

With the usual definition , only 1 and 3435 fulfill the property of a Munchausen number. ${\displaystyle 0^{0}=1}$

Narcissistic numbers have a similar law of formation , but there the potencies are fixed (in different ways). In the case of Münchhausen numbers, however, each individual digit determines the number with which it is raised to the power. Since, in a figurative sense, each digit “pulls itself up”, the Münchhausen numbers got their name from an allusion to a well-known tale of lies by the Baron von Münchhausen .

See also

• Happy number
• Lucky number
• Fortunate number
• Narcissistic number

Web links

• Follow A046253 in OEIS
• Eric W. Weisstein : Münchhausen Number . In: MathWorld (English).

literature

• Daan van Berkel: On a curious property of 3435. ( PDF )