Apronic number is a number that is the product of two consecutiveintegers, that is, a number of the form${\displaystyle n(n+1)}$.^[1] The study of these numbers dates back toAristotle. They are also calledoblong numbers,heteromecic numbers,^[2] orrectangular numbers;^[3] however, the term "rectangular number" has also been applied to thecomposite numbers.^[4][5]

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... (sequenceA002378 in theOEIS).

Letting${\displaystyle P_n}$ denote the pronic number${\displaystyle n(n+1)}$, we have${\displaystyle P_{-n}=P_{n-1}}$. Therefore, in discussing pronic numbers, we may assume that${\displaystyle n\ge 0}$without loss of generality, a convention that is adopted in the following sections.

The pronic numbers were studied asfigurate numbers alongside thetriangular numbers andsquare numbers inAristotle'sMetaphysics,^[2] and their discovery has been attributed much earlier to thePythagoreans.^[3]As a kind of figurate number, the pronic numbers are sometimes calledoblong^[2] because they are analogous topolygonal numbers in this way:^[1]

1 × 2	2 × 3	3 × 4	4 × 5

Thenth pronic number is the sum of the firstneven integers, and as such is twice thenth triangular number^[1][2] andn more than thenthsquare number, as given by the alternative formulan² +n for pronic numbers. Hence thenth pronic number and thenth square number (the sum of thefirstn odd integers) form asuperparticular ratio:

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

Due to this ratio, thenth pronic number is at aradius ofn andn + 1 from a perfect square, and thenth perfect square is at a radius ofn from a pronic number. Thenth pronic number is also the difference between theodd square(2n + 1)² and the(n+1)stcentered hexagonal number.

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

The partial sum of the firstn positive pronic numbers is twice the value of thenthtetrahedral number:

${\displaystyle \sum _{k=1}^{n}k(k+1)=\frac{n(n+1)(n+2)}{3}=2T_n}$.

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

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

Thepartial sum of the firstn terms in this series is^[7]

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

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

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

Pronic numbers are even, and 2 is the onlyprime pronic number. It is also the only pronic number in theFibonacci sequence and the only pronicLucas number.^[8][9]

Thearithmetic mean of two consecutive pronic numbers is asquare number:

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

So there is a square between any two consecutive pronic numbers. It is unique, since

Another consequence of this chain of inequalities is the following property. Ifm is a pronic number, then the following holds:

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

The fact that consecutive integers arecoprime 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 factorsn orn + 1. Thus a pronic number issquarefree if and only ifn andn + 1 are also squarefree. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors ofn andn + 1.

If 25 is appended to thedecimal 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

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

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

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

• Integer sequences
• Figurate numbers

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