A heptagonal number is a figurate number

Figurate number

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...

 that represents a heptagon. The n-th heptagonal number is given by the formula.

The first few heptagonal numbers are:

1, 7, 18

18 (number)

18 is the natural number following 17 and preceding 19.In speech, the numbers 18 and 80 are sometimes confused. When carefully enunciated, they differ in which syllable is stressed: 18 vs 80 . However, in dates such as 1864, or when contrasting numbers in the teens, such as 17, 18, 19, the stress...

, 34

34 (number)

34 is the natural number following 33 and preceding 35.-In mathematics:34 is the ninth distinct semiprime and has four divisors including unity and itself. Its neighbors, 33 and 35, also are distinct semiprimes, having four divisors each, and 34 is the smallest number to be surrounded by numbers...

, 55

55 (number)

55 is the natural number following 54 and preceding 56.-Albania:*Gazeta 55, a newspaper*Constitution law 55, a law during Communist Albania.-Mathematics:...

, 81

81 (number)

81 is the natural number following 80 and preceding 82.-In mathematics:Eighty-one is the square of 9 and the fourth power of 3. Like all powers of three, 81 is a perfect totient number. It is a heptagonal number and a centered octagonal number. It is also a tribonacci number, and an open meandric...

, 112

112 (number)

112 is the natural number following 111 and preceding 113.-In mathematics:One hundred [and] twelve is an abundant number and a heptagonal number...

, 148

148 (number)

148 is the natural number following 147 and before 149.-In mathematics:* 148 is the second number to be both a heptagonal number and a centered heptagonal number * 148 is the twelfth member of the Mian–Chowla sequence...

, 189

189 (number)

189 is the natural number following 188 and preceding 190.-In mathematics:* 189 is an odd number* 189 is a centered cube number* 189 is a composite number* 189 is a deficient number, as 131 is less than 189* 189 is a heptagonal number...

, 235

235 (number)

235 is the integer after 234 and before 236.-In mathematics:It is the second number of the tenth row of a skewed Fibonacci-Pascal triangle ....

, 286, 342, 403, 469, 540, 616

616 (number)

616 is the natural number following 615 and preceding 617.-In mathematics:616 is the 25th member of the Padovan sequence, coming after 265, 351, 465 . 616 is a polygonal number in four different ways: it is a heptagonal number, as well as 13-, 31- and 104-gonal.It is also the sum of the squares of...

, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, …

The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square number

Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...

s, the digital root

Digital root

The digital root of a number is the value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum...

 in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number

Triangular number

A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

.

A generalized heptagonal number is obtained by the formula
where T_n is the nth triangular number. The first few generalized heptagonal numbers are:

1, 4, 7, 13

13 (number)

13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....

, 18, 27

27 (number)

27 is the natural number following 26 and preceding 28.- In mathematics :Twenty-seven is a perfect cube, being 33 = 3 × 3 × 3. 27 is also 23 . There are exactly 27 straight lines on a smooth cubic surface, which give a basis of the fundamental representation of the E6 Lie algebra...

, 34, 46

46 (number)

46 is the natural number following 45 and preceding 47.- In mathematics :Forty-six is a Wedderburn-Etherington number, an enneagonal number and a centered triangular number. It is the sum of the totient function for the first twelve integers. 46 is the largest even integer that can't be expressed...

, 55, 70

70 (number)

70 is the natural number following 69 and preceding 71.-In mathematics:Its factorization makes it a sphenic number. 70 is a Pell number and a generalized heptagonal number, one of only two numbers to be both. Also, it is the seventh pentagonal number and the fourth 13-gonal number, as well as the...

, 81, 99

99 (number)

99 is the natural number following 98 and preceding 100.-Mathematics:99 is the ninth repdigit, a palindromic number and a Kaprekar number...

, 112, …

Every other generalized heptagonal number is a regular heptagonal number. Besides 1 and 70, no generalized heptagonal numbers are also Pell number

Pell number

In mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers...

s.
A formula for the sum of the reciprocals of the heptagonal numbers is given by: