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 ).

See also

• Regular heptagonal number.