Thenth star number is given by the formulaSn = 6n(n − 1) + 1. The first 45 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, 11353, and 11881. (sequenceA003154 in theOEIS)
Thedigital 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 (i.e., 13, 37 and 73) are also consecutive star numbers.
Proof without words that then-th star number is 12 times the (n−1)-th triangular number, plus one
Geometrically, thenth star number is made up of a central point and 12 copies of the (n−1)thtriangular number — making it numerically equal to thenthcentered dodecagonal number, but differently arranged. As such, the formula thenth star number can be written as S_n=1+12T_n-1 where T_n=n(n+1)/2.
Infinitely many star numbers are alsotriangular numbers, the first four beingS1 = 1 =T1,S7 = 253 =T22,S91 = 49141 =T313, andS1261 = 9533161 =T4366 (sequenceA156712 in theOEIS).
Infinitely many star numbers are alsosquare numbers, the first four beingS1 = 12,S5 = 121 = 112,S45 = 11881 = 1092, andS441 = 1164241 = 10792 (sequenceA054318 in theOEIS), for square stars (sequenceA006061 in theOEIS).
Astar prime is a star number that isprime. The first few star primes (sequenceA083577 in theOEIS) are13, 37, 73, 181, 337, 433, 541, 661, 937.
Asuperstar prime is a star prime whose prime index is also a star number. The first two such numbers are 661 and 1750255921.
Areverse 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 tooctagonal numbers.[1]
