Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Star number
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{short description|Centered figurate number}} {{Infobox integer sequence | image = Star number 73 as sum of gnomons.svg | image_size = 200px | alt = | caption = First four star numbers, by color. | number = [[infinity]] | formula = <math>S_n=6n(n-1)+1</math> | first_terms = [[1 (number)|1]], [[13 (number)|13]], [[37 (number)|37]], [[73 (number)|73]], [[121 (number)|121]], [[181 (number)|181]] | OEIS = A003154 | OEIS_name = star }} [[File:Chinese checkers start.svg|thumb|The Chinese checkers board has 121 holes.]] In [[mathematics]], a '''star number''' is a [[centered number|centered]] [[figurate number]], a centered [[hexagram]] (six-pointed star), such as the [[Star of David]], or the board [[Chinese checkers]] is played on. {| style="line-height: 1em; min-width: 290px;" cellspacing="10" ! 1 !! !! 13 !! !! 37 |- align="center" valign="middle" |[[Image:RedDotX.svg|16px|*]] | |[[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] | |[[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:GrayDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]][[Image:RedDotX.svg|16px|*]]<br>[[Image:RedDotX.svg|16px|*]] |} The ''n''th star number is given by the formula ''S<sub>n</sub>'' = 6''n''(''n'' β 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. {{OEIS|id=A003154}} 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. == Relationships to other kinds of numbers == [[File:Star_number_visual_proof.svg|thumb|[[Proof without words]] that the ''n''-th star number is 12 times the (''n''−1)-th triangular number, plus one]] Geometrically, the ''n''th star number is made up of a central point and 12 copies of the (''n''β1)th [[triangular number]] β making it numerically equal to the ''n''th [[centered polygonal number|centered dodecagonal number]], but differently arranged. As such, the formula the ''n''th star number can be written as S_n=1+12T_n-1 where T_n=n(n+1)/2. Infinitely many star numbers are also [[triangular number]]s, the first four being ''S''<sub>1</sub> = 1 = ''T''<sub>1</sub>, ''S''<sub>7</sub> = 253 = ''T''<sub>22</sub>, ''S''<sub>91</sub> = 49141 = ''T''<sub>313</sub>, and ''S''<sub>1261</sub> = 9533161 = ''T''<sub>4366</sub> {{OEIS|id=A156712}}. Infinitely many star numbers are also [[square number]]s, the first four being ''S''<sub>1</sub> = 1<sup>2</sup>, ''S''<sub>5</sub> = 121 = 11<sup>2</sup>, ''S''<sub>45</sub> = 11881 = 109<sup>2</sup>, and ''S''<sub>441</sub> = 1164241 = 1079<sup>2</sup> {{OEIS|id=A054318}}, for square stars {{OEIS|id=A006061}}. A '''star prime''' is a star number that is [[prime number|prime]]. The first few star primes {{OEIS|id=A083577}} 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]]s.<ref>{{cite OEIS|A000567}}</ref> ==Other properties== The [[Harmonic series (mathematics)|harmonic series]] of [[unit fraction]]s with the star numbers as denominators is: <math display=block> \begin{align} \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\sqrt3}\tan (\frac \pi {2\sqrt3})\\ &\approx 1.159173.\\ \end{align} </math> The [[alternating series]] of [[unit fraction]]s with the star numbers as denominators is: <math display=block> \begin{align} \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{align} </math> ==See also== * [[Centered hexagonal number]] == References == <references/> {{Figurate numbers}} {{Classes of natural numbers}} [[Category:Figurate numbers]] [[Category:Star of David]]
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)
Pages transcluded onto the current version of this page
(
help
)
:
Template:Cite OEIS
(
edit
)
Template:Classes of natural numbers
(
edit
)
Template:Figurate numbers
(
edit
)
Template:Infobox integer sequence
(
edit
)
Template:OEIS
(
edit
)
Template:Short description
(
edit
)