181 (number)

181 is prime, and a palindromic,{{Cite OEIS |A002385 |Palindromic primes: prime numbers whose decimal expansion is a palindrome. |access-date=2023-11-02 }} strobogrammatic,{{Cite OEIS |

A007597 |Strobogrammatic primes. |access-date=2023-11-02 }} and dihedral number{{Cite OEIS |A134996 |Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes. |access-date=2023-11-02 }} in decimal. 181 is a Chen prime.{{Cite OEIS |A109611 |Chen primes: primes p such that p + 2 is either a prime or a semiprime. |access-date=2016-05-26 }}

181 is a twin prime with 179,{{Cite OEIS |A006512 |Greater of twin primes. |access-date=2023-11-02 }} equal to the sum of five consecutive prime numbers:{{Cite OEIS |A034964 |Sums of five consecutive primes. |access-date=2023-11-02 }} 29 + 31 + 37 + 41 + 43.

181 is the difference of two consecutive square numbers 91² – 90²,{{Cite OEIS |A024352 |Numbers which are the difference of two positive squares, c^2 - b^2 with 1 less than or equal to b less than c. |access-date=2023-11-02 }} as well as the sum of two consecutive squares: 9² + 10².{{Cite OEIS |A001844 |Centered square numbers: a(n) equal to 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z is Y+1) ordered by increasing Z; then sequence gives Z values. |access-date=2016-05-26}}

As a centered polygonal number,{{Cite web |editor=Sloane, N. J. A. |editor-link=Neil Sloane |title=Centered polygonal numbers |website=The On-Line Encyclopedia of Integer Sequences |url=https://oeis.org/wiki/Centered_polygonal_numbers |publisher=OEIS Foundation |access-date=2023-11-02 }} 181 is:

{{Bulletlist

|a centered square number,

|a centered pentagonal number,{{Cite OEIS |A005891 |Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net. |access-date=2016-05-26 }}

|a centered dodecagonal number,{{Cite OEIS |A003154 |Centered 12-gonal numbers. Also star numbers: 6*n*(n-1) + 1. |access-date=2016-05-26}}

|a centered 18-gonal number,{{Cite OEIS |A069131 |Centered 18-gonal numbers. |access-date=2016-05-26}} and

|a centered 30-gonal number.

}}

181 is also a centered (hexagram) star number, as in the game of Chinese checkers.

Specifically, 181 is the 42nd prime number{{Cite OEIS |A000040 |The prime numbers |access-date=2023-11-02 }} and 16th full reptend prime in decimal,{{Cite OEIS |A001913 |Full reptend primes: primes with primitive root 10. |access-date=2023-11-02 }} where multiples of its reciprocal $\backslash tfrac\; \{1\}\{181\}$ inside a prime reciprocal magic square repeat 180 digits with a magic sum $M$ of 810; this value is one less than 811, the 141st prime number and 49th full reptend prime (or equivalently long prime) in decimal whose reciprocal repeats 810 digits. While the first full non-normal prime reciprocal magic square is based on $\backslash tfrac\; \{1\}\{19\}$ with a magic constant of 81 from a $18\; \backslash times\; 18$ square,{{Cite book|last=Andrews |first=William Symes |title=Magic Squares and Cubes |url=http://djm.cc/library/Magic_Squares_Cubes_Andrews_edited.pdf |publisher=Open Court Publishing Company |location=Chicago, IL |year=1917 |pages=176, 177 |isbn=9780486206585 |oclc=1136401 |zbl=1003.05500 |mr=0114763 }} a normal $19\; \backslash times\; 19$ magic square has a magic constant $M\_\{19\}\; =\; 19\; \backslash times\; 181$;{{Cite OEIS |A006003 |a(n) equal to n*(n^2 + 1)/2. |access-date=2023-11-02 }} the next such full, prime reciprocal magic square is based on multiples of the reciprocal of 383 (also palindromic).{{Cite OEIS |A072359 |Primes p such that the p-1 digits of the decimal expansion of k/p (for k equal to 1,2,3,...,p-1) fit into the k-th row of a magic square grid of order p-1. |access-date=2023-09-04 }}{{efn|1=Where the full reptend index of 181 is 16 = 4², the such index of 811 is 49 = 7². Note, also, that 282 is 141 × 2. }}

181 is an undulating number in ternary and nonary numeral systems, while in decimal it is the 28th undulating prime.{{Cite OEIS |A032758 |Undulating primes (digits alternate). |access-date=2023-11-02 }}

==References==

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• [http://primes.utm.edu/curios/page.php/181.html Prime curiosities: 181]
• [https://web.archive.org/web/20061002152926/http://athensohio.net/reference/number/181/ Number Facts and Trivia: 181]
• [http://www.numbergossip.com/181 Number Gossip: 181]

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Category:Integers