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 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).

If $n$ is a pronic number, then the following is true:

\lfloor {\sqrt {n}}\rfloor \cdot \lceil {\sqrt {n}}\rceil =n

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]


1 × 2	2 × 3	3 × 4	4 × 5

The $n$ th pronic number 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. 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.

Sum of pronic numbers

The sum of the reciprocals of the pronic numbers (excluding 0) is a telescoping series that sums to 1:^[6]

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

The partial sum of the first $n$ terms in this series is^[6]

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

The partial sum of the first $n$ 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}={\frac {n^{\overline {3}}}{3}}.

Additional properties

The $n$ th pronic number is the sum of the first $n$ even integers, and as such are twice the $n$ th triangular number.^[2] All 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.^[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 = 25², 1225 = 35². This is because

(10n+5)^{2}=100n^{2}+100n+25=100n(n+1)+25\,

.

A related conjecture

It is conjectured that every non-pronic number >4 can be written as n(n+1)+p, where n is a nonnegative integer (n(n+1) is a pronic number, including 0), p is either an odd prime (when this number is odd) or twice an odd prime (when this number is even), e.g.

5 = 0*1+5 = 1*2+3

7 = 0*1+7 = 1*2+5

8 = 1*2+2*3

9 = 1*2+7 = 2*3+3

10 = 0*1+2*5

11 = 0*1+11 = 2*3+5

13 = 0*1+13 = 1*2+11 = 2*3+7

14 = 0*1+2*7

15 = 1*2+13 = 3*4+3

16 = 1*2+2*7 = 2*3+2*5

17 = 0*1+17 = 2*3+11 = 3*4+5

18 = 3*4+2*3

19 = 0*1+19 = 1*2+17 = 2*3+13 = 3*4+7

21 = 1*2+19

22 = 0*1+2*11 = 3*4+2*5

23 = 0*1+23 = 2*3+17 = 3*4+11 = 4*5+3

24 = 1*2+2*11

25 = 1*2+23 = 3*4+13 = 4*5+5

Note that there are some pronic numbers which cannot be written as this form, such as 72 and 210, it is conjectured that every non-pronic number >4 can be written as this form.

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. (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, 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, MR 1605341.
^ Rummel, Rudolf J. (1988), Applied Factor Analysis, Northwestern University Press, p. 319, ISBN 9780810108240.

[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.

[telescope-6] 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.

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

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

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

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v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect