عدد نجمي

عدد نجمي

First four star numbers, by color.
إجمالي عدد المصطلحات	infinity
المعادلة	$S_{n}=6n(n-1)+1$
أول المصطلحات	1, 13, 37, 73, 121, 181
OEIS index	A003154

The Chinese checkers board has 121 holes.

العدد النجمي star number هو عدد شكلي ممركز يمثل شكل نجمة سداسية hexagram، أو the board Chinese checkers is played on.

1		13		37

The nth star number is given by the formula S_n = 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 (المتتالية A003154 في OEIS)

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 (i.e., 13, 37 and 73) are also consecutive star numbers.

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 Relationships to other kinds of numbers

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.

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 (المتتالية A156712 في OEIS).

Infinitely many star numbers are also square numbers, the first four being S₁ = 1², S₅ = 121 = 11², S₄₅ = 11881 = 109², and S₄₄₁ = 1164241 = 1079² (المتتالية A054318 في OEIS), for square stars (المتتالية A006061 في OEIS).

A star prime is a star number that is prime. The first few star primes (المتتالية A083577 في OEIS) 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 numbers.

 Other properties

The harmonic series of unit fractions with the star numbers as denominators is:

$${\begin{aligned}\sum _{n=1}^{\infty }&{\frac {1}{S_{n}}}\\&=1+{\frac {1}{13}}+{\frac {1}{37}}+{\frac {1}{73}}+{\frac {1}{121}}+{\frac {1}{181}}+{\frac {1}{253}}+{\frac {1}{337}}+\cdots \\&={\frac {\pi }{2{\sqrt {3}}}}\tan({\frac {\pi }{2{\sqrt {3}}}})\\&\approx 1.159173.\\\end{aligned}}$$

The alternating series of unit fractions with the star numbers as denominators is:

$${\begin{aligned}\sum _{n=1}^{\infty }&(-1)^{n-1}{\frac {1}{S_{n}}}\\&=1-{\frac {1}{13}}+{\frac {1}{37}}-{\frac {1}{73}}+{\frac {1}{121}}-{\frac {1}{181}}+{\frac {1}{253}}-{\frac {1}{337}}+\cdots \\&\approx 0.941419.\\\end{aligned}}$$

تعطى صيغة العدد النجمي للعدد n بالعلاقة: 6n(n - 1) + 1

تعطى الأعداد الأولى من سلسلة الأعداد النجمية كالتالي: 1 - 13 - 37 - 73 - 121 - 181 - 253 - 337 - 433 - 541 - 661 - 793 - 937 - ...

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