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A pronic number is a number that is the product of two consecutive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, that is, a number of the form n(n+1).. The study of these numbers dates back to
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
. They are also called oblong numbers, heteromecic numbers,. or rectangular numbers; however, the term "rectangular number" has also been applied to the
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s. The first few pronic numbers are: : 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110,
132 132 may refer to: *132 (number) *AD 132 *132 BC __NOTOC__ Year 132 BC was a year of the Roman calendar, pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Laenas and Rupilius (or, less frequently, year 622 ''Ab ...
, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 … . Letting P_n denote the pronic number n(n+1), we have P_ = P_. Therefore, in discussing pronic numbers, we may assume that n\geq 0 without loss of generality, a convention that is adopted in the following sections.


As figurate numbers

The pronic numbers were studied as figurate numbers alongside the
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
s and
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
s in
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
's ''
Metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
'', and their discovery has been attributed much earlier to the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
.. As a kind of figurate number, the pronic numbers are sometimes called ''oblong'' because they are analogous to polygonal numbers in this way: : The th pronic number is the sum of the first even integers, and as such is twice the th triangular number and more than the th
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
, as given by the alternative formula for pronic numbers. The th pronic number is also the difference between the odd square and the st centered hexagonal number. Since the number of off-diagonal entries in a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
is twice a triangular number, it is a pronic number.


Sum of pronic numbers

The partial sum of the first positive pronic numbers is twice the value of the th tetrahedral number: :\sum_^ k(k+1) =\frac= 2T_n. The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1:. :\sum_^ \frac=\frac+\frac+\frac\cdots=1. The partial sum of the first terms in this series is :\sum_^ \frac =\frac.


Additional properties

Pronic numbers are even, and 2 is the only
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
pronic number. It is also the only pronic number in the
Fibonacci sequence In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
and the only pronic Lucas number.. The
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...
of two consecutive pronic numbers is a
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
: :\frac = (n+1)^2 So there is a square between any two consecutive pronic numbers. It is unique, since :n^2 \leq n(n+1) < (n+1)^2 < (n+1)(n+2) < (n+2)^2. Another consequence of this chain of inequalities is the following property. If is a pronic number, then the following holds: : \lfloor\rfloor \cdot \lceil\rceil = m. The fact that consecutive integers are
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
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 or . Thus a pronic number is squarefree if and only if and are also squarefree. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of and . If 25 is appended to the
decimal representation A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, ...
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 :100n(n+1) + 25 = 100n^2 + 100n + 25 = (10n+5)^2\,.


References

{{Classes of natural numbers Integer sequences Figurate numbers