pronic number

A pronic number is a number that is the product of two consecutive integers, that is, a number of the form $n(n+1)$ .{{citation |first1=J. H. |last1=Conway |author1-link=John H. Conway |first2=R. K. |last2=Guy |author2-link=Richard K. Guy |title=The Book of Numbers |location=New York |publisher=Copernicus |at=Figure 2.15, p. 34 |year=1996}}. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,{{citation

| last = Knorr | first = Wilbur Richard | author-link = Wilbur Knorr

| isbn = 90-277-0509-7

| location = Dordrecht-Boston, Mass.

| mr = 0472300

| pages = 144–150

| publisher = D. Reidel Publishing Co.

| title = The evolution of the Euclidean elements

| url = https://books.google.com/books?id=_1H6BwAAQBAJ&pg=PA144

| year = 1975}}. or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers.{{citation|url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2008.01.0238:section=42|title=Plutarch, De Iside et Osiride, section 42|website=www.perseus.tufts.edu|access-date=16 April 2018}}{{citation|title=Number Story: From Counting to Cryptography|first=Peter Michael|last=Higgins|publisher=Copernicus Books|year=2008|isbn=9781848000018|page=9|url=https://books.google.com/books?id=HcIwkWXy3CwC&pg=PA9}}.

The first 60 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, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... {{OEIS|id=A002378}}.

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

File:Illustration of Triangular Number Leading to a Rectangle.svg

File:Illustration of Pronic Number n^2 and nplus1^2.svg and {{mvar|n}}+1 less than the ({{mvar|n}}+1)st square]]

The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's Metaphysics, and their discovery has been attributed much earlier to the Pythagoreans.{{citation|title=Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1|series=Springer reference|first=Ari|last=Ben-Menahem|publisher=Springer-Verlag|year=2009|isbn=9783540688310|page=161|url=https://books.google.com/books?id=9tUrarQYhKMC&pg=PA161}}.

As a kind of figurate number, the pronic numbers are sometimes called oblong because they are analogous to polygonal numbers in this way:

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1 × 2

2 × 3

3 × 4

4 × 5

The {{mvar|n}}th pronic number is the sum of the first {{mvar|n}} even integers, and as such is twice the {{Mvar|n}}th triangular number and {{mvar|n}} more than the {{mvar|n}}th square number, as given by the alternative formula {{math|n² + n}} for pronic numbers. Hence the {{mvar|n}}th pronic number and the {{mvar|n}}th square number (the sum of the Square_number#Properties) form a superparticular ratio:

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

Due to this ratio, the {{mvar|n}}th pronic number is at a radius of {{mvar|n}} and {{mvar|n}} + 1 from a perfect square, and the {{mvar|n}}th perfect square is at a radius of {{mvar|n}} from a pronic number. The {{mvar|n}}th pronic number is also the difference between the odd square {{math|(2n + 1)²}} and the {{math|(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.{{citation|title=Applied Factor Analysis|first=Rudolf J.|last=Rummel|publisher=Northwestern University Press|year=1988|isbn=9780810108240|page=319|url=https://books.google.com/books?id=g_eNa_XzyEIC&pg=PA319}}.

Sum of pronic numbers

The partial sum of the first {{mvar|n}} positive pronic numbers is twice the value of the {{mvar|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:{{citation |last=Frantz |first=Marc |title=The Calculus Collection: A Resource for AP and Beyond |pages=467–468 |year=2010 |editor1-last=Diefenderfer |editor1-first=Caren L. |series=Classroom Resource Materials |contribution=The telescoping series in perspective |contribution-url=https://books.google.com/books?id=SHJ39945R1kC&pg=PA467 |publisher=Mathematical Association of America |isbn=9780883857618 |editor2-last=Nelsen |editor2-first=Roger B. |editor1-link=Caren Diefenderfer}}.

: $\sum_{i=1}^{\infty} \frac{1}{i(i+1)}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}\cdots=1$ .

The partial sum of the first {{mvar|n}} terms in this series is

: $\sum_{i=1}^{n} \frac{1}{i(i+1)} =\frac{n}{n+1}$ .

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.{{citation

| last = McDaniel

| first = Wayne L.

| issue = 1

| journal = Fibonacci Quarterly

| mr = 1605345

| pages = 60–62

| title = Pronic Lucas numbers

| url = http://www.mathstat.dal.ca/FQ/Scanned/36-1/mcdaniel2.pdf

| volume = 36

| year = 1998

| doi = 10.1080/00150517.1998.12428962

| access-date = 2011-05-21

| archive-url = https://web.archive.org/web/20170705130526/http://www.mathstat.dal.ca/FQ/Scanned/36-1/mcdaniel2.pdf

| archive-date = 2017-07-05

| url-status = dead

}}.{{citation

| last = McDaniel | first = Wayne L.

| issue = 1

| journal = Fibonacci Quarterly

| mr = 1605341

| pages = 56–59

| title = Pronic Fibonacci numbers

| url = http://www.fq.math.ca/Scanned/36-1/mcdaniel1.pdf

| volume = 36

| year = 1998| doi = 10.1080/00150517.1998.12428961

}}.

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 {{mvar|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 {{mvar|n}} or {{math|n + 1}}. Thus a pronic number is squarefree if and only if {{mvar|n}} and {{math|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 {{mvar|n}} and {{math|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:{{cite web | last=Meyer | first=David | title=A Useful Mathematical Trick, Telescoping Series, and the Infinite Sum of the Reciprocals of the Triangular Numbers | url=https://davidmeyer.github.io/qc/tricks.pdf | page=1 | website=David Meyer's GitHub | access-date=2024-11-26 }}

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

References

Category:Integer sequences

Category:Figurate numbers