Star number


A star number is a centered figurate number a centered hexagram, such as the one that Chinese checkers is played on.
The nth star number is given by the formula Sn = 6n + 1. The first 43 star numbers are
1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837.
The digital root of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93.
Unique among the star numbers is 35113, since its prime factors are also consecutive star numbers.

Relationships to other kinds of numbers

Geometrically, the nth star number is made up of a central point and 12 copies of the 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.
Infinitely many star numbers are also square numbers, the first four being S1 = 12, S5 = 121 = 112, S45 = 11881 = 1092, and S441 = 1164241 = 10792.
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.
The term "star number" or "stellate number" is occasionally used to refer to octagonal numbers.