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A star number is a
centered 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 (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
* polygon ...
, a centered
hexagram
, can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green).
A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
(six-pointed star), such as the Star of David, or the board
Chinese checkers is played on.
The ''n''th star number is given by the formula ''S
n'' = 6''n''(''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
The
digital root
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit su ...
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.
Relationships to other kinds of numbers
Geometrically, the ''n''th star number is made up of a central point and 12 copies of the (''n''−1)th
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
— making it numerically equal to the ''n''th
centered dodecagonal number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side i ...
, but differently arranged.
Infinitely many star numbers are also
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
s, the first four being ''S''
1 = 1 = ''T''
1, ''S''
7 = 253 = ''T''
22, ''S''
91 = 49141 = ''T''
313, and ''S''
1261 = 9533161 = ''T''
4366 .
Infinitely many star numbers are also
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s, the first four being ''S''
1 = 1
2, ''S''
5 = 121 = 11
2, ''S''
45 = 11881 = 109
2, and ''S''
441 = 1164241 = 1079
2 , for square stars .
A star prime is a star number that is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. 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 number
An octagonal number is a figurate number that represents an octagon. The octagonal number for ''n'' is given by the formula 3''n''2 - 2''n'', with ''n'' > 0. The first few octagonal numbers are
: 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 34 ...
s.
Other properties
The
harmonic series of
unit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...
s with the star numbers as denominators is:
The
alternating series
In mathematics, an alternating series is an infinite series of the form
\sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n
with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternatin ...
of
unit fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc ...
s with the star numbers as denominators is:
See also
*
Centered hexagonal number
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following ...
References
{{Classes of natural numbers
Figurate numbers