# Centered heptagonal number

A centered heptagonal number is a centered figurate number 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

${\displaystyle {7n^{2}-7n+2} \over 2}$.

The first few centered heptagonal numbers are

1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953 (sequence A069099 in the OEIS)

## Properties

• Centered heptagonal numbers alternate parity in the pattern odd-even-even-odd.
• A heptagonal numbers can expressed as a multiple of a triangular number by 7, plus one:
${\displaystyle C_{7,n}=7*T_{n-1}+1}$
• ${\displaystyle C_{7,n}}$ is the sum of the integers between n+1 and 3n+1 (including) minus the sum of the integers from 0 to n (including).

## Centered heptagonal prime

A centered heptagonal prime is a centered heptagonal number that is prime. The first few centered heptagonal primes are

43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, ... (sequence A144974 in the OEIS)

Due to parity, the centered heptagonal primes are in the form of ${\displaystyle C_{7,(4n)}}$ or ${\displaystyle C_{7,(4n+1)}}$.

The centered heptagonal twin prime numbers are

43, 71, 197, 463, 1933, 5741, 8233, 9283, 11173, 14561, 34651, ... (sequence A144975 in the OEIS).