Pronic number

Demonstration, with Cuisenaire rods, of pronic numbers n=1, n=2, and n=3 (2, 6, and 12).

A pronic number is a number which is the product of two consecutive integers, that is, a number of the form Template:Math. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the "rectangular number" name has also been applied to the composite numbers.

The first few 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 … Template:OEIS.

If n is a pronic number, then the following is true:

$\lfloor {\sqrt {n}}\rfloor \cdot \lceil {\sqrt {n}}\rceil =n$

As figurate numbers

Template:Math = Template:Math

The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's Metaphysics, and their discovery has been attributed much earlier to the Pythagoreans. As a kind of figurate number, the pronic numbers are sometimes called oblong because they are analogous to polygonal numbers in this way:

1×2	2×3	3×4	4×5

The Template:Mvarth pronic number is twice the Template:Mvarth triangular number and Template:Mvar more than the Template:Mvarth square number, as given by the alternative formula Template:Math for pronic numbers. The Template:Mvarth pronic number is also the difference between the odd square Template:Math and the Template:Mathst centered hexagonal number.

Sum of pronic numbers

The sum of the reciprocals of the pronic numbers (excluding 0) is a telescoping series that sums to 1:

$1={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{12}}\cdots =\sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}.$

The partial sum of the first Template:Mvar terms in this series is

$\sum _{i=1}^{n}{\frac {1}{i(i+1)}}={\frac {n}{n+1}}.$

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

$\sum _{k=1}^{n}k(k+1)={\frac {n(n+1)(n+2)}{3}}=2T_{n}={\frac {n^{\overline {3}}}{3}}.$

Additional properties

First four pronic numbers as sums of the first n even numbers.

The Template:Mvarth pronic number is the sum of the first Template:Mvar even integers. All 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.

The number of off-diagonal entries in a square matrix is always a pronic number.

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 n or n+1. 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 e.g. 625 = 25², 1225 = 35². This is because

$(10n+5)^{2}=100n^{2}+100n+25=100n(n+1)+25\,$.