111 (number)

111 is the fourth non-trivial nonagonal number,{{cite OEIS|1=A001106|2=9-gonal (or enneagonal or nonagonal) numbers|access-date=2016-05-26}} and the seventh perfect totient number.{{Cite OEIS|1=A082897|2=Perfect totient numbers|access-date=2016-05-26}}

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

:$\backslash varphi(111)\; =\; \backslash varphi(\backslash sigma(111)).${{Cite OEIS |A006872 |Numbers k such that phi(k) is phi(sigma(k)). |access-date=2024-02-03 }}

Two other of its multiples (333 and 555) also have the same property (with totients of 216 and 288, respectively).{{efn|1=Also,{{Bullet list |The 111st composite number 146{{Cite OEIS |A002808 |The composite numbers: numbers n of the form x*y for x > 1 and y > 1. |access-date=2024-02-03 }} 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}, members which add to 146 (without including 9). |357, in turn the index of 444 as a composite, is the twentieth such number, following 333. |The composite index of 1000 is 831, 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. }}

File:Smallest perfect squared squares.svg 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:{{Cite book |author=Henry E. Dudeney |author-link=Henry Dudeney |title=Amusements in Mathematics |publisher=Thomas Nelson & Sons, Ltd. |location=London |year=1917 |page=125 |url=http://djm.cc/library/Amusements_in_Mathematics_Dudeney_edited02.pdf |isbn=978-1153585316 |oclc=645667320 }}

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

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).{{efn|1=Relatedly, 111 is also the magic constant of the n-Queens Problem for n = 6.{{Cite OEIS|1=A006003|2=a(n) = n*(n^2 + 1)/2}} }}

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).{{Cite journal |last=Gambini |first=Ian |title=A method for cutting squares into distinct squares |journal=Discrete Applied Mathematics |volume=98 |issue=1–2 |publisher=Elsevier |location=Amsterdam |pages=65–80 |year=1999 |doi=10.1016/S0166-218X(99)00158-4 |doi-access=free |mr=1723687 |zbl=0935.05024 }}

111 is $R\_\{3\}$ or the second repunit in decimal,{{cite OEIS|A002275|Repunits: (10^n - 1)/9. Often denoted by R_n.}} 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 $3n\; \backslash times\; 37$. 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) {{OEIS|id=A002384}}. In base 10, it is furthermore a strobogrammatic number,{{Cite OEIS|1=A000787|2=Strobogrammatic numbers|access-date=2022-05-07}} as well as a Harshad number.{{Cite OEIS|1=A005349|2=Niven (or Harshad) numbers|access-date=2016-05-26}}

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

==Nelson==

{{Main|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:

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

{{Notelist}}

{{Reflist}}

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

{{Commons category}}

{{Integers|1}}

{{DEFAULTSORT:111 (Number)}}

Category:Horatio Nelson

Category:Integers