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# polite number

A polite number $n$ is an integer that is the sum of two or more consecutive nonnegative integers in at least one way. To put it algebraically, if $n$ is polite then there is a solution to

$n=\sum_{{i=a}}^{b}i$ |

with $b>a$ and $a>-1$. For example, 42 is a polite number since it is the sum of the integers from 3 to 9. The first few polite numbers are 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, etc.

Obviously all triangular numbers are polite numbers. So are all odd numbers. In fact, the numbers that are not polite are the powers of 2.

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## Mathematics Subject Classification

11A25*no label found*

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## Attached Articles

all positive integers are polite numbers except powers of two by PrimeFan

another proof that a number is polite iff it is positive and not a positive power of $2$ by CWoo

proof of all positive integers are polite numbers except powers of two by n847530

table of polite number representations for $1 < n < 101$ by PrimeFan

another proof that a number is polite iff it is positive and not a positive power of $2$ by CWoo

proof of all positive integers are polite numbers except powers of two by n847530

table of polite number representations for $1 < n < 101$ by PrimeFan