Innumber theory, apolite number is apositive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is calledimpolite.^[1][2] The impolite numbers are exactly thepowers of two, and the polite numbers are thenatural numbers that are not powers of two.

Polite numbers have also been calledstaircase numbers because theYoung diagrams which represent graphically thepartitions of a polite number into consecutive integers (in theFrench notation of drawing these diagrams) resemblestaircases.^[3][4][5] If all numbers in the sum are strictly greater than one, the numbers so formed are also calledtrapezoidal numbers because they represent patterns of points arranged in atrapezoid.^{[6][7][8][9][10][11][12]}

The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied bySylvester,^[13] Mason,^[14][15]Leveque,^[16] and many other more recent authors.^{[1][2][17][18][19][20][21][22][23]} The polite numbers describe the possible numbers of sides of theReinhardt polygons.^[24]

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, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, ... (sequenceA138591 in theOEIS).

The impolite numbers are exactly thepowers of two.^[13] It follows from theLambek–Moser theorem that thenth polite number isf(n + 1), where

${\displaystyle f(n)=n+\lfloor {\log }_2\left(n+{\log }_{2}n\right)\rfloor .}$

Thepoliteness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For everyx, the politeness ofx equals the number ofodddivisors ofx that are greater than one.^[13] The politeness of the numbers 1, 2, 3, ... is

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 3, ... (sequenceA069283 in theOEIS).

For instance, the politeness of 9 is 2 because it has two odd divisors, 3 and 9, and two polite representations

9 = 2 + 3 + 4 = 4 + 5;

the politeness of 15 is 3 because it has three odd divisors, 3, 5, and 15, and (as is familiar tocribbage players)^[25] three polite representations

15 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 = 7 + 8.

An easy way of calculating the politeness of a positive number by decomposing the number into itsprime factors, taking the powers of all prime factors greater than 2, adding 1 to all of them, multiplying the numbers thus obtained with each other and subtracting 1. For instance 90 has politeness 5 because${\displaystyle 90=2\times 3^2\times 5^1}$; the powers of 3 and 5 are respectively 2 and 1, and applying this method${\displaystyle (2+1)\times (1+1)-1=5}$.

To see the connection between odd divisors and polite representations, suppose a numberx has the odd divisory > 1. Theny consecutive integers centered onx/y (so that their average value isx/y) havex as their sum:

${\displaystyle x=\sum _{i=\frac{x}{y}-\frac{y-1}{2}}^{\frac{x}{y}+\frac{y-1}{2}}i.}$

Some of the terms in this sum may be zero or negative. However, if a term is zero it can be omitted and any negative terms may be used to cancel positive ones, leading to a polite representation forx. (The requirement thaty > 1 corresponds to the requirement that a polite representation have more than one term; applying the same construction fory = 1 would just lead to the trivial one-term representationx = x.)For instance, the polite numberx = 14 has a single nontrivial odd divisor, 7. It is therefore the sum of 7 consecutive numbers centered at 14/7 = 2:

14 = (2 − 3) + (2 − 2) + (2 − 1) + 2 + (2 + 1) + (2 + 2) + (2 + 3).

The first term, −1, cancels a later +1, and the second term, zero, can be omitted, leading to the polite representation

14 = 2 + (2 + 1) + (2 + 2) + (2 + 3) = 2 + 3 + 4 + 5.

Conversely, every polite representation ofx can be formed from this construction. If a representation has an odd number of terms,x/y is the middle term, while if it has aneven number of terms and its minimum value ism it may be extended in a unique way to a longer sequence with the same sum and an odd number of terms, by including the 2m − 1 numbers −(m − 1), −(m − 2), ..., −1, 0, 1, ...,m − 2,m − 1.After this extension, again,x/y is the middle term. By this construction, the polite representations of a number and its odd divisors greater than one may be placed into aone-to-one correspondence, giving abijective proof of the characterization of polite numbers and politeness.^[13][26] More generally, the same idea gives a two-to-one correspondence between, on the one hand, representations as a sum of consecutive integers (allowing zero, negative numbers, and single-term representations) and on the other hand odd divisors (including 1).^[15]

Another generalization of this result states that, for anyn, the number of partitions ofn into odd numbers havingk distinct values equals the number of partitions ofn into distinct numbers havingk maximal runs of consecutive numbers.^[13][27][28] Here a run is one or more consecutive values such that the next larger and the next smaller consecutive values are not part of the partition; for instance the partition 10 = 1 + 4 + 5 has two runs, 1 and 4 + 5.A polite representation has a single run, and a partition with one valued is equivalent to a factorization ofn as the productd ⋅ (n/d), so the special casek = 1 of this result states again the equivalence between polite representations and odd factors (including in this case the trivial representationn = n and the trivial odd factor 1).

If a polite representation starts with 1, the number so represented is atriangular number

${\displaystyle T_n=\frac{n(n+1)}{2}=1+2+\cdots +n.}$

Otherwise, it is the difference of two nonconsecutive triangular numbers

${\displaystyle i+(i+1)+(i+2)+\cdots +j=T_j-T_{i-1}(j>i\ge 2).}$

This second case is called a trapezoidal number.^[12] One can also consider polite numbers that aren't trapezoidal. The only such numbers are the triangular numbers with only one nontrivial odd divisor, because for those numbers, according to thebijection described earlier, the odd divisor corresponds to the triangular representation and there can be no other polite representations. Thus, non-trapezoidal polite number must have the form of a power of two multiplied by an odd prime. As Jones and Lord observe,^[12] there are exactly two types of triangular numbers with this form:

1. the evenperfect numbers 2^n − 1(2ⁿ − 1) formed by the product of aMersenne prime 2ⁿ − 1 with half the nearestpower of two, and
2. the products 2^n − 1(2ⁿ + 1) of aFermat prime 2ⁿ + 1 with half the nearest power of two.

(sequenceA068195 in theOEIS). For instance, the perfect number 28 = 2^3 − 1(2³ − 1) and the number 136 = 2^4 − 1(2⁴ + 1) are both this type of polite number. It is conjectured that there are infinitely many Mersenne primes, in which case there are also infinitely many polite numbers of this type.

• Polite Numbers, NRICH, University of Cambridge, December 2002
• Introducing Runsums, R. Knott.
• Is there any pattern to the set of trapezoidal numbers? Intellectualism.org question of the day, October 2, 2003. With a diagram showing trapezoidal numbers color-coded by the number of terms in their expansions.

• Additive number theory
• Figurate numbers
• Integer sequences
• Quadrilaterals

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