A star number is a centered

Centered number

The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides...

 figurate number

Figurate number

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...

 that represents a centered hexagram, such as the one that Chinese checkers

Chinese checkers

Chinese checkers is a board game that can be played by two, three, four, or six people, playing individually or with partners...

 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

13 (number)

13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....

, 37

37 (number)

37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...

, 73

73 (number)

73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...

, 121

121 (number)

121 is the natural number following 120 and preceding 122.-In mathematics:One hundred [and] twenty-one is a square and is the sum of three consecutive primes . There are no squares besides 121 known to be of the form 1 + p + p^2 + p^3 + p^4, where p is prime...

, 181

181 (number)

181 is the natural number between 180 and 182.-In mathematics:* 181 is an odd number* 181 is a centered number** 181 is a centered pentagonal number** 181 is a centered 12-gonal number** 181 is a centered 18-gonal number...

, 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 .

Geometrically, the nth star number is made up of a central point and 12 copies of the (n-1)th triangular number

Triangular number

A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

 — making it numerically equal to the nth centered dodecagonal number, but differently arranged.

The digital root

Digital root

The digital root of a number is the value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum...

 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 number

Triangular number

A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

s. 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 . These are the values n=(x+2)/4 with x an even solution of the Diophantine equation

Diophantine equation

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

 

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 . These n values are n=(y+1)/2 from the Diophantine equation

Diophantine equation

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

 

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

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

. The first few star primes are

13, 37, 73, 181, 337, 433, 541, 661, 937.