Pronic number
Encyclopedia
A pronic number, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive integers, that is, n (n + 1). The n-th pronic number is twice the n-th triangular number
and n more than the n-th square number. The first few pronic numbers are:
These numbers are sometimes called oblong because they are analogous to polygonal number
s in this way:
Pronic numbers can also be expressed as n² + n. The n-th pronic number is the sum of the first n even integers, as well as the difference between (2n − 1)² and the n-th centered hexagonal number
.
All pronic numbers are even
, therefore 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 its factors. Thus a pronic number is squarefree
if and only if n and n + 1 are. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n + 1.
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
and n more than the n-th square number. The first few pronic numbers are:
- 00 (number)0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
, 2, 6, 1212 (number)12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
, 2020 (number)20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
, 3030 (number)30 is the natural number following 29 and preceding 31.-In mathematics:30 is the sum of the first four squares, which makes it a square pyramidal number.It is a primorial and is the smallest Giuga number....
, 4242 (number)42 is the natural number immediately following 41 and directly preceding 43. The number has received considerable attention in popular culture as a result of its central appearance in The Hitchhiker's Guide to the Galaxy as the "Answer to the Ultimate Question of Life, the Universe, and...
, 5656 (number)56 is the natural number following 55 and preceding 57.- Mathematics :56 is the sum of the first six triangular numbers , as well as the sum of six consecutive primes . It is also a tetranacci number and a pronic number. Adding up the divisors of 1 through 8 gives 56...
, 7272 (number)72 is the natural number following 71 and preceding 73. It is half a gross or 6 dozen .-In mathematics:...
, 9090 (number)90 is the natural number preceded by 89 and followed by 91.- In mathematics :*Because 90 is the sum of its unitary divisors , it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number. 90 is a pronic number. But it is also a...
, 110110 (number)110 is the natural number following 109 and preceding 111.It is also known as "eleventy", a term made famous by linguist and author J. R. R...
, 132132 (number)132 is the natural number following 131 and preceding 133.-In mathematics:132 is the sixth Catalan number. It is a pronic number, the product of 11 and 12. As it has 12 divisors total, 132 is a refactorable number....
, 156, 182, 210, 240, 272, 306, 342, 380, 420420 (number)420 is the natural number following 419 and preceding 421.420 is the sum of four consecutive primes . 420 is a zero of the Mertens function and is sparsely totient....
, 462 …
These numbers are sometimes called oblong because they are analogous to polygonal number
Polygonal number
In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots were thought of as alphas . These are one type of 2-dimensional figurate numbers.- Definition and examples :...
s in this way:
 |
|
|
|
| 1×2 | 2×3 | 3×4 | 4×5 |
Pronic numbers can also be expressed as n² + n. The n-th pronic number is the sum of the first n even integers, as well as the difference between (2n − 1)² and the n-th centered hexagonal number
Centered hexagonal number
A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice....
.
All pronic numbers are even
Even and odd numbers
In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
, therefore 2 is the only prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
pronic number. It is also the only pronic number in the Fibonacci sequence
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
and the only pronic Lucas number
Lucas number
The Lucas numbers are an integer sequence named after the mathematician François Édouard Anatole Lucas , who studied both that sequence and the closely related Fibonacci numbers...
.
The number of off-diagonal entries in a square matrix is always a pronic number.
The fact that consecutive integers are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
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 its factors. Thus a pronic number is squarefree
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
if and only if n and n + 1 are. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n + 1.