A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.
| 1 | 13 | 37 | ||
|---|---|---|---|---|
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The nth star number is given by the formula 6n(n - 1) + 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 (sequence A003154 in OEIS).
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.
The digital root of a star number is always 1 or 4. 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.
Not many star numbers are also triangular numbers. 1 and 253 are the only two such numbers in the list given above, corresponding to n=1 and n=7. There are infinitely many with the next two correspond to n=91 and n=1261 (sequence A003154 in OEIS). These are the values n=(x+2)/4 with x an even solution of the Diophantine equation x2 = 3y2 + 1.
Not many star numbers are also square. 1 and 121 are the only two such numbers in the list given above, corresponding to n=1 and n=5. There are infinitely many with the next two being n=45 and n=441 (sequence A054318 in OEIS). These n values are n=(y+1)/2 from the Diophantine equation 2x2 + 1 = 3y2.
Confusingly, the term "star number" or "stellate number" is occasionally used to refer to octagonal numbers.
A star prime is a star number that is prime. The first few star primes (sequence A083577 in OEIS) are
13, 37, 73, 181, 337, 433, 541, 661, 937.
See also
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