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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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Oct 7

new question: Lorenz system by David Bankom

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new correction: examples and OEIS sequences by fizzie

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new correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

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new question: Latent variable by adam_reith

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

## 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