111 (number)

In mathematics

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).

Magic squares

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

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==

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.

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

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

References

References

1. {{cite OEIS
2. {{Cite OEIS
3. {{Cite OEIS. A006872. Numbers k such that phi(k) is phi(sigma(k)).
4. {{Cite OEIS. A002808. The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
5. Henry E. Dudeney. (1917). "Amusements in Mathematics". [[Thomas Nelson (publisher).
6. {{Cite OEIS
7. Gambini, Ian. (1999). "A method for cutting squares into distinct squares". [[Elsevier]].
8. {{cite OEIS. A002275. Repunits: (10^n - 1)/9. Often denoted by R_n.
9. {{Cite OEIS
10. {{Cite OEIS