Polite number
In
Polite numbers have also been called staircase numbers because the
The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied by Sylvester,[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 the Reinhardt polygons.[24]
Examples and characterization
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, ... (sequence A138591 in the OEIS).
The impolite numbers are exactly the powers of two.[13] It follows from the Lambek–Moser theorem that the nth polite number is f(n + 1), where
Politeness
The politeness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For every x, the politeness of x equals the number of
The politeness of the numbers 1, 2, 3, ... isFor 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 to cribbage 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 its
Construction of polite representations from odd divisors
To see the connection between odd divisors and polite representations, suppose a number x has the odd divisor y > 1. Then y consecutive integers centered on x/y (so that their average value is x/y) have x as their sum:
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 for x. (The requirement that y > 1 corresponds to the requirement that a polite representation have more than one term; applying the same construction for y = 1 would just lead to the trivial one-term representation x = x.) For instance, the polite number x = 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 of x 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 an even number of terms and its minimum value is m 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 a one-to-one correspondence, giving a bijective 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 any n, the number of partitions of n into odd numbers having k distinct values equals the number of partitions of n into distinct numbers having k 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 value d is equivalent to a factorization of n as the product d ⋅ (n/d), so the special case k = 1 of this result states again the equivalence between polite representations and odd factors (including in this case the trivial representation n = n and the trivial odd factor 1).
Trapezoidal numbers
If a polite representation starts with 1, the number so represented is a triangular number
Otherwise, it is the difference of two nonconsecutive triangular numbers
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 the bijection 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:
- the even perfect numbers 2n − 1(2n − 1) formed by the product of a Mersenne prime 2n − 1 with half the nearest power of two, and
- the products 2n − 1(2n + 1) of a Fermat prime2n + 1 with half the nearest power of two.
(sequence A068195 in the OEIS). For instance, the perfect number 28 = 23 − 1(23 − 1) and the number 136 = 24 − 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.
References
- ^ S2CID 171530924.
- ^ S2CID 171681914.
- ISBN 978-0-201-10238-3.
- ^ Stacey, K.; Groves, S. (1985), Strategies for Problem Solving, Melbourne: Latitude.
- ^ Stacey, K.; Scott, N. (2000), "Orientation to deep structure when trying examples: a key to successful problem solving", in Carillo, J.; Contreras, L. C. (eds.), Resolucion de Problemas en los Albores del Siglo XXI: Una vision Internacional desde Multiples Perspectivas y Niveles Educativos (PDF), Huelva, Spain: Hergue, pp. 119–147, archived from the original (PDF) on 2008-07-26.
- JSTOR 2689901.
- ^ Jean, Charles-É. (March 1991), "Les nombres trapézoïdaux" (French), Bulletin de l'AMQ: 6–11.
- .
- .
- ^ Smith, Jim (1997), "Trapezoidal numbers", Mathematics in School, 5: 42.
- Bibcode:1999JIntS...2...16V, Article 99.1.6.
- ^ S2CID 125545112.
- ^ , H. F. Baker, ed. Sylvester defines the class of a partition into distinct integers as the number of blocks of consecutive integers in the partition, so in his notation a polite partition is of first class.
- ^ Mason, T. E. (1911), "On the representations of a number as a sum of consecutive integers", Proceedings of the Indiana Academy of Science: 273–274.
- ^ MR 1517654.
- S2CID 124093945,
- S2CID 14169613.
- MR 2187932.
- MR 1573264.
- Zbl 0475.10014.
- S2CID 125202845.
- .
- ^ Parker, John (1998), "Sums of consecutive integers", Mathematics in School, 27 (2): 8–11.
- MR 2793611
- ISBN 978-0-201-14236-5.
- ISBN 978-0-88385-806-6.
- MR 0202617.
- MR 0304323.
External links
- 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.