181 (number)

From HandWiki
Short description: Natural number
← 180 181 182 →
Cardinalone hundred eighty-one
(one hundred eighty-first)
Divisors1, 181
Greek numeralΡΠΑ´
Roman numeralCLXXXI
Base 365136

181 (one hundred [and] eighty-one) is the natural number following 180 and preceding 182.

In mathematics

181 is prime, and a palindromic,[1] strobogrammatic,[2] and dihedral number[3] in decimal. 181 is a Chen prime.[4]

181 is a twin prime with 179,[5] equal to the sum of five consecutive prime numbers:[6] 29 + 31 + 37 + 41 + 43.

181 is the difference of two consecutive square numbers 912 – 902,[7] as well as the sum of two consecutive squares: 92 + 102.[8]

As a centered polygonal number,[9] 181 is:

181 is also a centered (hexagram) star number,[11] as in the game of Chinese checkers.

Specifically, 181 is the 42nd prime number[13] and 16th full reptend prime in decimal,[14] where multiples of its reciprocal [math]\displaystyle{ \tfrac {1}{181} }[/math] inside a prime reciprocal magic square repeat 180 digits with a magic sum [math]\displaystyle{ M }[/math] 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 [math]\displaystyle{ \tfrac {1}{19} }[/math] with a magic constant of 81 from a [math]\displaystyle{ 18 \times 18 }[/math] square,[15] a normal [math]\displaystyle{ 19 \times 19 }[/math] magic square has a magic constant [math]\displaystyle{ M_{19} = 19 \times 181 }[/math];[16] the next such full, prime reciprocal magic square is based on multiples of the reciprocal of 383 (also palindromic).[17][lower-alpha 1]

181 is an undulating number in ternary and nonary numeral systems, while in decimal it is the 28th undulating prime.[18]

In other fields

181 is also:

See also

  • The year AD 181 or 181 BC
  • List of highways numbered 181
  • United Nations Security Council Resolution 181


  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.
  1. Sloane, N. J. A., ed. "Sequence A002385 (Palindromic primes: prime numbers whose decimal expansion is a palindrome.)". OEIS Foundation. https://oeis.org/A002385. Retrieved 2023-11-02. 
  2. Sloane, N. J. A., ed. "Sequence A007597 (Strobogrammatic primes.)". OEIS Foundation. https://oeis.org/A007597. Retrieved 2023-11-02. 
  3. Sloane, N. J. A., ed. "Sequence A134996 (Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes.)". OEIS Foundation. https://oeis.org/A134996. Retrieved 2023-11-02. 
  4. Sloane, N. J. A., ed. "Sequence A109611 (Chen primes: primes p such that p + 2 is either a prime or a semiprime.)". OEIS Foundation. https://oeis.org/A109611. Retrieved 2016-05-26. 
  5. Sloane, N. J. A., ed. "Sequence A006512 (Greater of twin primes.)". OEIS Foundation. https://oeis.org/A006512. Retrieved 2023-11-02. 
  6. Sloane, N. J. A., ed. "Sequence A034964 (Sums of five consecutive primes.)". OEIS Foundation. https://oeis.org/A034964. Retrieved 2023-11-02. 
  7. Sloane, N. J. A., ed. "Sequence 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.)". OEIS Foundation. https://oeis.org/A024352. Retrieved 2023-11-02. 
  8. 8.0 8.1 Sloane, N. J. A., ed. "Sequence 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.)". OEIS Foundation. https://oeis.org/A001844. Retrieved 2016-05-26. 
  9. 9.0 9.1 Sloane, N. J. A., ed. "Centered polygonal numbers". OEIS Foundation. https://oeis.org/wiki/Centered_polygonal_numbers. 
  10. Sloane, N. J. A., ed. "Sequence A005891 (Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for planar net.)". OEIS Foundation. https://oeis.org/A005891. Retrieved 2016-05-26. 
  11. 11.0 11.1 Sloane, N. J. A., ed. "Sequence A003154 (Centered 12-gonal numbers. Also star numbers: 6*n*(n-1) + 1.)". OEIS Foundation. https://oeis.org/A003154. Retrieved 2016-05-26. 
  12. Sloane, N. J. A., ed. "Sequence A069131 (Centered 18-gonal numbers.)". OEIS Foundation. https://oeis.org/A069131. Retrieved 2016-05-26. 
  13. Sloane, N. J. A., ed. "Sequence A000040 (The prime numbers)". OEIS Foundation. https://oeis.org/A000040. Retrieved 2023-11-02. 
  14. Sloane, N. J. A., ed. "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". OEIS Foundation. https://oeis.org/A001913. Retrieved 2023-11-02. 
  15. Andrews, William Symes (1917). Magic Squares and Cubes. Chicago, IL: Open Court Publishing Company. pp. 176, 177. ISBN 9780486206585. OCLC 1136401. http://djm.cc/library/Magic_Squares_Cubes_Andrews_edited.pdf. 
  16. Sloane, N. J. A., ed. "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". OEIS Foundation. https://oeis.org/A006003. Retrieved 2023-11-02. 
  17. Sloane, N. J. A., ed. "Sequence 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.)". OEIS Foundation. https://oeis.org/A072359. Retrieved 2023-09-04. 
  18. Sloane, N. J. A., ed. "Sequence A032758 (Undulating primes (digits alternate).)". OEIS Foundation. https://oeis.org/A032758. Retrieved 2023-11-02. 

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