pronic number
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A pronic number is a number which is the product of two consecutive
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s, that is, a number of the form .. The study of these numbers dates back to
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

Aristotle
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...
s. The first few pronic numbers are: :, , , , 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420 (number), 420, 462 … . If is a pronic number, then the following is true: : \lfloor\rfloor \cdot \lceil\rceil = n


As figurate numbers

The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

Aristotle
's ''Metaphysics (Aristotle), 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: : The arithmetic mean for two consecutive pronic numbers is a square number: : \frac = (n+1)^2 So between the consecutive pronic numbers there is always a square, and only unique one (since n^2 < n(n+1) < (n+1)^2 < (n+1)(n+2) < (n+2)^2 ). The th pronic number is twice the th triangular number and more than the th square number, as given by the alternative formula for pronic numbers. The th pronic number is also the difference between the Even and odd numbers, odd square and the st 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+\frac+\frac\cdots=\sum_^ \frac. The partial sum of the first terms in this series is :\sum_^ \frac =\frac. The partial sum of the first pronic numbers is twice the value of the th tetrahedral number: :\sum_^ k(k+1) =\frac= 2T_n =\frac.


Additional properties

The th pronic number is the sum of the first Even and odd numbers, even integers, and as such is twice the th triangular number. All pronic numbers are even, and 2 is the only prime number, prime pronic number. It is also the only pronic number in the Fibonacci number, 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 or . Thus a pronic number is Square-free integer, 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 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\,.


References

{{Classes of natural numbers Integer sequences Figurate numbers