111 is the fourth non-trivial
nonagonal number, and the seventh
perfect totient number. 111 is furthermore the ninth number such that its
Euler totient \varphi(n) of
72 is equal to the totient value of its
sum-of-divisors: :\varphi(111) = \varphi(\sigma(111)). Two other of its multiples (
333 and
555) also have the same property (with totients of
216 and
288, respectively).{{efn|1=Also, 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}, 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). 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).
Properties in certain radices 111 is R_{3} or the second
repunit in
decimal, 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 \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) . In base 10, it is furthermore a
strobogrammatic number, as well as a
Harshad number. In
base 18, the number 111 is 73 (= 34310) which is the only base where 111 is a
perfect power. ==Nelson== 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. 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==