Star number

Geometrically, the nth star number is made up of a central point and 12 copies of the (n−1)th triangular number — making it numerically equal to the nth centered dodecagonal number, but differently arranged.

Infinitely many star numbers are also triangular numbers, the first four being S₁ = 1 = T₁, S₇ = 253 = T₂₂, S₉₁ = 49141 = T₃₁₃, and S₁₂₆₁ = 9533161 = T₄₃₆₆ (sequence A156712 in the OEIS).

Infinitely many star numbers are also square numbers, the first four being S₁ = 1², S₅ = 121 = 11², S₄₅ = 11881 = 109², and S₄₄₁ = 1164241 = 1079² (sequence A054318 in the OEIS), for square stars (sequence A006061 in the OEIS).

A star prime is a star number that is prime. The first few star primes (sequence A083577 in the OEIS) are 13, 37, 73, 181, 337, 433, 541, 661, 937.

A superstar prime is a star prime whose prime index is also a star number. The first two such numbers are 661 and 1750255921.

A reverse superstar prime is a star number whose index is a star prime. The first few such numbers are 937, 7993, 31537, 195481, 679393, 1122337, 1752841, 2617561, 5262193.

The term "star number" or "stellate number" is occasionally used to refer to octagonal numbers.[1]

The harmonic series of unit fractions with the star numbers as denominators is:

$${\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }&{\frac {1}{S_{n}}}\\&=1+{\frac {1}{13}}+{\frac {1}{37}}+{\frac {1}{73}}+{\frac {1}{121}}+{\frac {1}{181}}+{\frac {1}{253}}+{\frac {1}{337}}+\cdots \\&={\frac {\pi }{2{\sqrt {3}}}}\tan({\frac {\pi }{2{\sqrt {3}}}})\\&\approx 1.159173.\\\end{aligned}}}$$

The alternating series of unit fractions with the star numbers as denominators is:

$${\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }&(-1)^{n-1}{\frac {1}{S_{n}}}\\&=1-{\frac {1}{13}}+{\frac {1}{37}}-{\frac {1}{73}}+{\frac {1}{121}}-{\frac {1}{181}}+{\frac {1}{253}}-{\frac {1}{337}}+\cdots \\&\approx 0.941419.\\\end{aligned}}}$$

• Centered hexagonal number