← 110 111 112 →
Cardinalone hundred eleven
Ordinal111th
(one hundred eleventh)
Factorization3 × 37
Divisors1, 3, 37, 111
Greek numeralΡΙΑ´
Roman numeralCXI
Binary11011112
Ternary110103
Senary3036
Octal1578
Duodecimal9312
Hexadecimal6F16

111 (one hundred [and] eleven) is the natural number following 110 and preceding 112.

In mathematics

111 is a perfect totient number.[1]

111 is R3 or the second repunit (in decimal), a number like 11, 111, or 1111 that consists of repeated units, or 1's. It equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are repdigits of the form 3n × 37. As a repunit, it also follows that 111 is a palindromic number.

All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in base ten, but is prime in base two, where 1112 = 710. It is also prime in these other bases up to 128: 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119 (sequence A002384 in the OEIS)

In base 18, the number 111 is 73 (= 34310) which is the only base where 111 is a perfect power.

111 is the ninth number such that its Euler totient (of 72) is equal to the totient value of its sum-of-divisors: ${\displaystyle \varphi (111)=\varphi (\sigma (111))}$.[2] Two other of its multiples (333 and 555) also have the same property (with totients of 216 and 288, respectively).[a]

The smallest magic square using only 1 and prime numbers has a magic constant of 111:[4]

 31 73 7 13 37 61 67 1 43

A six-by-six magic square using the numbers 1 to 36 also has a magic constant of 111:

 1 11 31 29 19 20 2 22 24 25 8 30 3 33 26 23 17 9 34 27 10 12 21 7 35 14 15 16 18 13 36 4 5 6 28 32

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).

111 is also the magic constant of the n-Queens Problem for n = 6.[5] It is also a nonagonal number.[6]

In base 10, it is a Harshad number[7] as well as a strobogrammatic number.[8]

Nelson

 Main article: Nelson (cricket)

In cricket, the number 111 is sometimes called "a Nelson" after Admiral Nelson, who allegedly only had "One Eye, One Arm, One Leg" near the end of his life. This is in fact inaccurate—Nelson never lost a leg. Alternate meanings include "One Eye, One Arm, One Ambition" and "One Eye, One Arm, One Arsehole".

Particularly in cricket, multiples of 111 are called a double Nelson (222), triple Nelson (333), quadruple Nelson (444; also known as a salamander) and so on.

A score of 111 is considered by some to be unlucky. To combat the supposed bad luck, some watching lift their feet off the ground. Since an umpire cannot sit down and raise his feet, the international umpire David Shepherd had a whole retinue of peculiar mannerisms if the score was ever a Nelson multiple. He would hop, shuffle, or jiggle, particularly if the number of wickets also matched—111/1, 222/2 etc.

111 is also:

Notes

1. ^ Also,[2]
• The 111st composite number 146[3] is the twelfth number whose totient value is the same value held by its sum-of-divisors.
• 357, in turn the index of 444 as a composite,[3] is the twentieth such number, following 333.
• The composite index of 1000 is 831,[3] the thirty-fifth member in this sequence of numbers to have a totient also shared by its sum-of-divisors, where 1000 is 1 + 999.
The only two numbers in decimal less than 1000 whose prime factorisations feature primes concatenated into a new prime are 138 and 777 (as 2 × 3 × 23 and 3 × 7 × 37, respectively), which add to 915. This sum represents the 38th member in the aforementioned sequence.[2]

References

1. ^ Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 26 May 2016.
2. ^ a b c Sloane, N. J. A. (ed.). "Sequence A006872 (Numbers k such that phi(k) is phi(sigma(k)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 3 February 2024.
3. ^ a b c Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 3 February 2024.
4. ^ Henry E. Dudeney (1917). Amusements in Mathematics (PDF). London: Thomas Nelson & Sons, Ltd. p. 125. ISBN 978-1153585316. OCLC 645667320.
5. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) = n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. ^ Sloane, N. J. A. (ed.). "Sequence A001106 (9-gonal (or enneagonal or nonagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 26 May 2016.
7. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 26 May 2016.
8. ^ Sloane, N. J. A. (ed.). "Sequence A000787 (Strobogrammatic numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 7 May 2022.
9. ^ John Ronald Reuel Tolkien (1993). The fellowship of the ring: being the first part of The lord of the rings. HarperCollins. ISBN 978-0-261-10235-4.

Further reading

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134