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 ) {\displaystyle 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 {\displaystyle P_{n}} denote the pronic number n ( n + 1 ) {\displaystyle n(n+1)} , we have P − n = P n − 1 {\displaystyle P_{{-}n}=P_{n{-}1}} . Therefore, in discussing pronic numbers, we may assume that n ≥ 0 {\displaystyle 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 nth pronic number is n more than the nth 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 nth pronic number is the sum of the first n even integers, and as such is twice the nth triangular number[1][2] and n more than the nth square number, as given by the alternative formula n2 + 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.

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 nth tetrahedral number:

∑ k = 1 n k ( k + 1 ) = n ( n + 1 ) ( n + 2 ) 3 = 2 T n {\displaystyle \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]

∑ i = 1 ∞ 1 i ( i + 1 ) = 1 2 + 1 6 + 1 12 + 1 20 ⋯ = 1 {\displaystyle \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]

∑ i = 1 n 1 i ( i + 1 ) = n n + 1 {\displaystyle \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:

∑ i = 1 ∞ ( − 1 ) i + 1 i ( i + 1 ) = 1 2 − 1 6 + 1 12 − 1 20 ⋯ = log ⁡ ( 4 ) − 1 {\displaystyle \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:

n ( n + 1 ) + ( n + 1 ) ( n + 2 ) 2 = ( n + 1 ) 2 {\displaystyle {\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 ≤ n ( n + 1 ) < ( n + 1 ) 2 < ( n + 1 ) ( n + 2 ) < ( n + 2 ) 2 . {\displaystyle 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:

⌊ m ⌋ ⋅ ⌈ m ⌉ = m . {\displaystyle \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

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

References

  1. ^ a b c Conway, J. H.; Guy, R. K. (1996), The Book of Numbers, New York: Copernicus, Figure 2.15, p. 34.
  2. ^ 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.
  3. ^ a b 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.
  7. ^ 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.
  8. ^ 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.
  9. ^ McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF), Fibonacci Quarterly, 36 (1): 56–59, MR 1605341.
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