Star number

Relationships to other kinds of numbers

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. As such, the formula the nth star number can be written as S_n=1+12T_n-1 where T_n=n(n+1)/2.

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 .

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

A star prime is a star number that is prime. The first few star primes 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.

Other properties

The harmonic series of unit fractions with the star numbers as denominators is: \begin{align} \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\sqrt3}\tan (\frac \pi {2\sqrt3})\ &\approx 1.159173.\ \end{align}

The alternating series of unit fractions with the star numbers as denominators is: \begin{align} \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{align}

References

References

1. {{cite OEIS. A000567