A centered heptagonal number is a centered

Centered number

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides...

 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 with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The centered heptagonal number for n is given by the formula
.

This can also be calculated by multiplying the 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...

 for (n - 1) by 7, then adding 1.

The first few centered heptagonal numbers are

1, 8, 22

22 (number)

22 is the natural number following 21 and preceding 23.- In mathematics :Twenty-two is an even composite number, its proper divisors being 1, 2 and 11....

, 43

43 (number)

43 is the natural number following 42 and preceding 44.- In mathematics :Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, and the next is forty-seven. 43 is the smallest prime that is not a Chen prime...

, 71

71 (number)

71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 106

106 (number)

106 is the natural number following 105 and preceding 107.-In mathematics:106 is the thirty-first distinct biprime and the fifteenth of the form...

, 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...

, 197

197 (number)

197 is the natural number following 196 and preceding 198.-In mathematics:* 197 is an odd number* 197 is a prime number** 197 is a Chen prime** 197 is an Eisenstein prime with no imaginary part** 197 is a strong prime** 197 is a twin prime with 199...

, 253, 316, 386, 463, 547, 638, 736, 841, 953

Centered heptagonal numbers alternate parity in the pattern odd-even-even-odd.

Centered heptagonal prime

A centered heptagonal prime is a centered heptagonal number that is prime

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. The first few centered heptagonal primes are

43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, ...

and centered heptagonal twin prime numbers are

43, 71, 197, 463, 1933, 5741, 8233, 9283, 11173, 14561, 34651, ... .