Pronic number

A pronic number is a number that 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 term "rectangular number" 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).

Letting $P_{n}$ denote the pronic number $n(n+1)$ , we have $P_{{-}n}=P_{n{-}1}$ . 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

Twice a triangular number is a pronic number

The $n$ th pronic number is $n$ more than the $n$ th square number

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]


1 × 2	2 × 3	3 × 4	4 × 5

The $n$ th pronic number is the sum of the first $n$ even integers, and as such is twice the $n$ th triangular number^[1]^[2] and $n$ more than the $n$ th square number, as given by the alternative formula $n 2 + n$ for pronic numbers. Hence the $n$ th pronic number and the $n$ th square number (the sum of the first $n$ odd integers) form a superparticular ratio:

{\frac {n(n+1)}{n^{2}}}={\frac {n+1}{n}}

Due to this ratio, the $n$ th pronic number is at a radius of $n$ and $n$ + 1 from a perfect square, and the $n$ th perfect square is at a radius of $n$ from a pronic number. The $n$ th pronic number is also the difference between the odd square $(2 n + 1) 2$ and the $(n +1)$ st centered hexagonal number.

Since the number of off-diagonal entries in a square matrix is twice a triangular number, it is a pronic number.^[6]

Sum of pronic numbers

The partial sum of the first $n$ positive pronic numbers is twice the value of the $n$ th tetrahedral number:

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

.

The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1:^[7]

\sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{20}}\cdots =1

.

The partial sum of the first $n$ terms in this series is^[7]

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

.

The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series:

\sum _{i=1}^{\infty }{\frac {(-1)^{i+1}}{i(i+1)}}={\frac {1}{2}}-{\frac {1}{6}}+{\frac {1}{12}}-{\frac {1}{20}}\cdots =\log(4)-1

.

Additional properties

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.^[8]^[9]

The arithmetic mean of two consecutive pronic numbers is a square number:

{\frac {n(n+1)+(n+1)(n+2)}{2}}=(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 $m$ is a pronic number, then the following holds:

\lfloor {\sqrt {m}}\rfloor \cdot \lceil {\sqrt {m}}\rceil =m.

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, 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}

.

The difference between two consecutive unit fractions is the reciprocal of a pronic number:^[10]

{\frac {1}{n}}-{\frac {1}{n+1}}={\frac {(n+1)-n}{n(n+1)}}={\frac {1}{n(n+1)}}

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 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.
^ Rummel, Rudolf J. (1988), Applied Factor Analysis, Northwestern University Press, p. 319, ISBN 9780810108240.
^ ^a ^b Frantz, Marc (2010), "The telescoping series in perspective", in Diefenderfer, Caren L.; Nelsen, Roger B. (eds.), 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, doi:10.1080/00150517.1998.12428962, MR 1605345, archived from the original (PDF) on 2017-07-05, retrieved 2011-05-21.
^ McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF), Fibonacci Quarterly, 36 (1): 56–59, doi:10.1080/00150517.1998.12428961, MR 1605341.
^ Meyer, David. "A Useful Mathematical Trick, Telescoping Series, and the Infinite Sum of the Reciprocals of the Triangular Numbers" (PDF). David Meyer's GitHub. p. 1. Retrieved 2024-11-26.

[bon-1] Conway, J. H.; Guy, R. K. (1996), The Book of Numbers, New York: Copernicus, Figure 2.15, p. 34.

[knorr-2] 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.

[hist-3] Ben-Menahem, Ari (2009), Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1, Springer reference, Springer-Verlag, p. 161, ISBN 9783540688310.

[4] "Plutarch, De Iside et Osiride, section 42", www.perseus.tufts.edu, retrieved 16 April 2018

[5] Higgins, Peter Michael (2008), Number Story: From Counting to Cryptography, Copernicus Books, p. 9, ISBN 9781848000018.

[6] Rummel, Rudolf J. (1988), Applied Factor Analysis, Northwestern University Press, p. 319, ISBN 9780810108240.

[telescope-7] Frantz, Marc (2010), "The telescoping series in perspective", in Diefenderfer, Caren L.; Nelsen, Roger B. (eds.), The Calculus Collection: A Resource for AP and Beyond, Classroom Resource Materials, Mathematical Association of America, pp. 467–468, ISBN 9780883857618.

[8] McDaniel, Wayne L. (1998), "Pronic Lucas numbers" (PDF), Fibonacci Quarterly, 36 (1): 60–62, doi:10.1080/00150517.1998.12428962, MR 1605345, archived from the original (PDF) on 2017-07-05, retrieved 2011-05-21.

[9] McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF), Fibonacci Quarterly, 36 (1): 56–59, doi:10.1080/00150517.1998.12428961, MR 1605341.

[Meyer2024-10] Meyer, David. "A Useful Mathematical Trick, Telescoping Series, and the Infinite Sum of the Reciprocals of the Triangular Numbers" (PDF). David Meyer's GitHub. p. 1. Retrieved 2024-11-26.

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