111 (number)

"Number 111" redirects here. For the 1919 film, see Number 111 (1919 film). For the 1938 film, see Number 111 (1938 film).

For other uses, see 111 (disambiguation).

Natural number

| ← 110  | 111 |  112 →

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← 110 111 112 113 114 115 116 117 118 119 →

* List of numbers
* Integers

← 0 100 200 300 400 500 600 700 800 900 →

Cardinal| one hundred eleven
Ordinal| 111th
(one hundred eleventh)
Factorization| 3 × 37
Divisors| 1, 3, 37, 111
Greek numeral| ΡΙΑ´
Roman numeral| CXI, cxi
Binary| 11011112
Ternary| 110103
Senary| 3036
Octal| 1578
Duodecimal| 9312
Hexadecimal| 6F16

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

## In mathematics

[edit]

111 is the fourth non-trivial nonagonal number,[1] and the seventh perfect totient number.[2]

111 is furthermore the ninth number such that its Euler totient φ ( n ) {\displaystyle \varphi (n)} of 72 is equal to the totient value of its sum-of-divisors:

φ ( 111 ) = φ ( σ ( 111 ) ) . {\displaystyle \varphi (111)=\varphi (\sigma (111)).} [3]

Two other of its multiples (333 and 555) also have the same property (with totients of 216 and 288, respectively).[a]

### Magic squares

[edit]

111 is adjacent to 110 and 112, the minimal side lengths of perfect squared squares that are tiled by smaller squares of distinct side lengths.

The smallest magic square using only 1 and prime numbers has a magic constant of 111:[5]

31 | 73 | 7

|
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13 | 37 | 61
67 | 1 | 43

Also, 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

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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).[b]

On the other hand, 111 lies between 110 and 112, which are the two smallest edge-lengths of squares that are tiled in the interior by smaller squares of distinct edge-lengths (see, squaring the square).[7]

### Properties in certain radices

[edit]

111 is  R 3 {\displaystyle R_{3}} or the second repunit in decimal,[8] a number like 11, 111, or 1111 that consists of repeated units, or ones. 111 equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are repdigits of the form  3 n × 37 {\displaystyle 3n\times 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 decimal, but is prime in base two, where 1112 = 710. It is also prime in many other bases up to 128 (3, 5, 6, ..., 119) (sequence A002384 in the OEIS). In base 10, it is furthermore a strobogrammatic number,[9] as well as a Harshad number.[10]

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

## Nelson

[edit]

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.

## In other fields

[edit]

111 is also:

* The emergency telephone number in New Zealand; see 111 (emergency telephone number).
* NHS 111, a medical helpline in England and Scotland.

## Notes

[edit]

1. ^ Also,[3]

* The 111st composite number 146[4] is the twelfth number whose totient value is the same value held by its sum-of-divisors. The sequence of nonagonal numbers that precede 111 is {0, 1, 9, 24, 46, 75},[1] members which add to 146 (without including 9).
* 357, in turn the index of 444 as a composite,[4] is the twentieth such number, following 333.
* The composite index of 1000 is 831,[4] 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.[3]
2. ^ Relatedly, 111 is also the magic constant of the n-Queens Problem for n = 6.[6]

## References

[edit]

1. ^ a b 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.
2. ^ Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 26 May 2016.
3. ^ 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.
4. ^ 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.
5. ^ Henry E. Dudeney (1917). Amusements in Mathematics (PDF). London: Thomas Nelson & Sons, Ltd. p. 125\. ISBN 978-1153585316. OCLC 645667320. `{{cite book}}`: ISBN / Date incompatibility (help)
6. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) = n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. ^ Gambini, Ian (1999). "A method for cutting squares into distinct squares". Discrete Applied Mathematics. 98 (1–2). Amsterdam: Elsevier: 65–80\. doi:10.1016/S0166-218X(99)00158-4. MR 1723687. Zbl 0935.05024.1–2&rft.pages=65-80&rft.date=1999&rft_id=https://zbmath.org/?format=complete&q;=an:0935.05024#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1723687#id-name=MR&rft_id=info:doi/10.1016/S0166-218X(99)00158-4&rft.aulast=Gambini&rft.aufirst=Ian&rft_id=https://doi.org/10.1016%2FS0166-218X%2899%2900158-4&rfr_id=info:sid/en.wikipedia.org:111+(number)" class="Z3988">
8. ^ Sloane, N. J. A. (ed.). "Sequence A002275 (Repunits: (10^n - 1)/9. Often denoted by R_n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
9. ^ Sloane, N. J. A. (ed.). "Sequence A000787 (Strobogrammatic numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 7 May 2022.
10. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 26 May 2016.

### Further reading

[edit]

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

## External links

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Wikimedia Commons has media related to 111 (number).

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