Pronic number

Template:Short description A pronic number is a number that is the product of two consecutive integers, that is, a number of the form <math>n(n+1)</math>.<ref name="bon">Template:Citation.</ref> The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,<ref name="knorr">Template:Citation.</ref> or rectangular numbers;<ref name="hist"/> however, the term "rectangular number" has also been applied to the composite numbers.<ref>Template:Citation</ref><ref>Template:Citation.</ref>

The first 60 pronic numbers are:

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... (sequence A002378 in the OEIS).

Letting <math>P_n</math> denote the pronic number <math>n(n+1)</math>, we have <math>P_{{-}n} = P_{n{-}1}</math>. Therefore, in discussing pronic numbers, we may assume that <math>n\geq 0</math> without loss of generality, a convention that is adopted in the following sections.

As figurate numbers

File:Illustration of Triangular Number Leading to a Rectangle.svg

Twice a triangular number is a pronic number

File:Illustration of Pronic Number n^2 and nplus1^2.svg

The Template:Mvarth pronic number is Template:Mvar more than the Template:Mvarth square number and Template:Mvar+1 less than the (Template:Mvar+1)st square

The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's Metaphysics,<ref name="knorr"/> and their discovery has been attributed much earlier to the Pythagoreans.<ref name="hist">Template:Citation.</ref> As a kind of figurate number, the pronic numbers are sometimes called oblong<ref name="knorr"/> because they are analogous to polygonal numbers in this way:<ref name="bon"/>

* *	* * * * * *	* * * * * * * * * * * *	* * * * * * * * * * * * * * * * * * * *
1 × 2	2 × 3	3 × 4	4 × 5

The Template:Mvarth pronic number is the sum of the first Template:Mvar even integers, and as such is twice the Template:Mvarth triangular number<ref name="bon"/><ref name="knorr"/> and Template:Mvar more than the Template:Mvarth square number, as given by the alternative formula Template:Math for pronic numbers. Hence the Template:Mvarth pronic number and the Template:Mvarth square number (the sum of the [[Square_number#Properties|first Template:Mvar odd integers]]) form a superparticular ratio:

<math>

\frac{n(n+1)}{n^2} = \frac{n + 1}{n} </math>

Due to this ratio, the Template:Mvarth pronic number is at a radius of Template:Mvar and Template:Mvar + 1 from a perfect square, and the Template:Mvarth perfect square is at a radius of Template:Mvar from a pronic number. The Template:Mvarth pronic number is also the difference between the odd square Template:Math and the Template:Mathst centered hexagonal number.

Since the number of off-diagonal entries in a square matrix is twice a triangular number, it is a pronic number.<ref>Template:Citation.</ref>

Sum of pronic numbers

The partial sum of the first Template:Mvar positive pronic numbers is twice the value of the Template:Mvarth tetrahedral number:

<math>\sum_{k=1}^{n} k(k+1) =\frac{n(n+1)(n+2)}{3}= 2T_n </math>.

The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1:<ref name="telescope">Template:Citation.</ref>

<math>\sum_{i=1}^{\infty} \frac{1}{i(i+1)}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}\cdots=1</math>.

The partial sum of the first Template:Mvar terms in this series is<ref name="telescope" />

<math>\sum_{i=1}^{n} \frac{1}{i(i+1)} =\frac{n}{n+1}</math>.

The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series:

<math>\sum_{i=1}^{\infty} \frac{(-1)^{i+1}}{i(i+1)}=\frac{1}{2}-\frac{1}{6}+\frac{1}{12}-\frac{1}{20}\cdots=\log(4)-1</math>.

Additional properties

Pronic numbers are even, and 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number.<ref>Template:Citation.</ref><ref>Template:Citation.</ref>

The arithmetic mean of two consecutive pronic numbers is a square number:

<math>\frac {n(n+1) + (n+1)(n+2)}{2} = (n+1)^2</math>

So there is a square between any two consecutive pronic numbers. It is unique, since

<math>n^2 \leq n(n+1) < (n+1)^2 < (n+1)(n+2) < (n+2)^2.</math>

Another consequence of this chain of inequalities is the following property. If Template:Mvar is a pronic number, then the following holds:

<math> \lfloor{\sqrt{m}}\rfloor \cdot \lceil{\sqrt{m}}\rceil = m.</math>

The fact that consecutive integers are coprime and that a pronic number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a pronic number is present in only one of the factors Template:Mvar or Template:Math. Thus a pronic number is squarefree if and only if Template:Mvar and Template:Math are also squarefree. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of Template:Mvar and Template:Math.

If 25 is appended to the decimal representation of any pronic number, the result is a square number, the square of a number ending on 5; for example, 625 = 25² and 1225 = 35². This is so because

<math>100n(n+1) + 25 = 100n^2 + 100n + 25 = (10n+5)^2</math>.

The difference between two consecutive unit fractions is the reciprocal of a pronic number:<ref name="Meyer2024">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>\frac{1}{n} - \frac{1}{n+1} = \frac{(n+1) - n}{n(n+1)} = \frac{1}{n(n+1)}</math>