181 (number)

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

← 180 181 182 →
← 180 181 182 183 184 185 186 187 188 189 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred eighty-one
Ordinal	181st (one hundred eighty-first)
Factorization	prime
Prime	42nd
Divisors	1, 181
Greek numeral	ΡΠΑ´
Roman numeral	CLXXXI, clxxxi
Binary	101101012
Ternary	202013
Senary	5016
Octal	2658
Duodecimal	13112
Hexadecimal	B516

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:

• a centered square number,^[8]
• a centered pentagonal number,^[10]
• a centered dodecagonal number,^[11]
• a centered 18-gonal number,^[12] and
• a centered 30-gonal number.^[9]

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 ${\displaystyle {\tfrac {1}{181}}}$ inside a prime reciprocal magic square repeat 180 digits with a magic sum ${\displaystyle M}$ 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 ${\displaystyle {\tfrac {1}{19}}}$ with a magic constant of 81 from a ${\displaystyle 18\times 18}$ square,^[15] a normal ${\displaystyle 19\times 19}$ magic square has a magic constant ${\displaystyle M_{19}=19\times 181}$;^[16] the next such full, prime reciprocal magic square is based on multiples of the reciprocal of 383 (also palindromic).^[17][a]

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

References

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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
2. Sloane, N. J. A. (ed.). "Sequence A007597 (Strobogrammatic primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
5. Sloane, N. J. A. (ed.). "Sequence A006512 (Greater of twin primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
6. Sloane, N. J. A. (ed.). "Sequence A034964 (Sums of five consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
8. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
9. Sloane, N. J. A. (ed.). "Centered polygonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
10. Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
11. Sloane, N. J. A. (ed.). "Sequence A003154 (Centered 12-gonal numbers. Also star numbers: 6*n*(n-1) + 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
12. Sloane, N. J. A. (ed.). "Sequence A069131 (Centered 18-gonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
13. Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
14. Sloane, N. J. A. (ed.). "Sequence A001913 (Full reptend primes: primes with primitive root 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
15. Andrews, William Symes (1917). Magic Squares and Cubes (PDF). Chicago, IL: Open Court Publishing Company. pp. 176, 177. ISBN 9780486206585. MR 0114763. OCLC 1136401. Zbl 1003.05500.
16. Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-04.
18. Sloane, N. J. A. (ed.). "Sequence A032758 (Undulating primes (digits alternate).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.

External links

• Prime curiosities: 181
• Number Facts and Trivia: 181
• Number Gossip: 181