Pronic number
A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n(n + 1).^{[1]} The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,^{[2]} or rectangular numbers;^{[3]} however, the "rectangular number" name has also been applied to the composite numbers.^{[4]}^{[5]}
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 … (sequence A002378 in the OEIS).
If n is a pronic number, then the following is true:
As figurate numbers
The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's Metaphysics,^{[2]} and their discovery has been attributed much earlier to the Pythagoreans.^{[3]} As a kind of figurate number, the pronic numbers are sometimes called oblong^{[2]} because they are analogous to polygonal numbers in this way:^{[1]}
The nth pronic number is twice the nth triangular number^{[1]}^{[2]} and n more than the nth square number, as given by the alternative formula n^{2} + n for pronic numbers. The nth pronic number is also the difference between the odd square (2n + 1)^{2} and the (n+1)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:^{[6]}
The partial sum of the first n terms in this series is^{[6]}
The partial sum of the first n pronic numbers is twice the value of the nth tetrahedral number:
Additional properties
The nth pronic number is the sum of the first n even integers.^{[2]} It follows that all pronic numbers are even, and that 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number.^{[7]}^{[8]}
The number of off-diagonal entries in a square matrix is always a pronic number.^{[9]}
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 n and n + 1 are also squarefree. 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.
If 25 is appended to the decimal representation of any pronic number, the result is a square number e.g. 625 = 25^{2}, 1225 = 35^{2}. This is because
- .
References
- ^ ^{a} ^{b} ^{c} Conway, J. H.; Guy, R. K. (1996), The Book of Numbers, New York: Copernicus, Figure 2.15, p. 34.
- ^ ^{a} ^{b} ^{c} ^{d} ^{e} Knorr, Wilbur Richard (1975), The evolution of the Euclidean elements, Dordrecht-Boston, Mass.: D. Reidel Publishing Co., pp. 144–150, ISBN 90-277-0509-7, MR 0472300.
- ^ ^{a} ^{b} Ben-Menahem, Ari (2009), Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1, Springer reference, Springer-Verlag, p. 161, ISBN 9783540688310.
- ^ "Plutarch, De Iside et Osiride, section 42". www.perseus.tufts.edu. Retrieved 16 April 2018.
- ^ Higgins, Peter Michael (2008), Number Story: From Counting to Cryptography, Copernicus Books, p. 9, ISBN 9781848000018.
- ^ ^{a} ^{b} Frantz, Marc (2010), "The telescoping series in perspective", in Diefenderfer, Caren L.; Nelsen, Roger B., The Calculus Collection: A Resource for AP and Beyond, Classroom Resource Materials, Mathematical Association of America, pp. 467–468, ISBN 9780883857618.
- ^ McDaniel, Wayne L. (1998), "Pronic Lucas numbers" (PDF), Fibonacci Quarterly, 36 (1): 60–62, MR 1605345.
- ^ McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF), Fibonacci Quarterly, 36 (1): 56–59, MR 1605341.
- ^ Rummel, Rudolf J. (1988), Applied Factor Analysis, Northwestern University Press, p. 319, ISBN 9780810108240.