111 (number)

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110 111 112
Cardinal one hundred eleven
Ordinal 111th
(one hundred eleventh)
Factorization 3 × 37
Divisors 1, 3, 37, 111
Greek numeral ΡΙΑ´
Roman numeral 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.

Contents

In mathematics

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 of 72 is equal to the totient value of its sum-of-divisors:

[3]

Two other of its multiples (333 and 555) also have the same property (with totients of 216 and 288, respectively). [lower-alpha 1]

Magic squares

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. Smallest perfect squared squares.svg
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]

31737
133761
67143

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

11131291920
2222425830
3332623179
34271012217
351415161813
364562832

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111). [lower-alpha 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). [7]

Properties in certain radices

111 is 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 . 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

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

111 is also:

See also

Notes

  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]

Related Research Articles

10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.

90 (ninety) is the natural number following 89 and preceding 91.

72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or 6 dozen.

34 (thirty-four) is the natural number following 33 and preceding 35.

58 (fifty-eight) is the natural number following 57 and preceding 59.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

300 is the natural number following 299 and preceding 301.

<span class="mw-page-title-main">360 (number)</span> Natural number

360 is the natural number following 359 and preceding 361.

144 is the natural number following 143 and preceding 145.

500 is the natural number following 499 and preceding 501.

800 is the natural number following 799 and preceding 801.

2000 is a natural number following 1999 and preceding 2001.

3000 is the natural number following 2999 and preceding 3001. It is the smallest number requiring thirteen letters in English.

4000 is the natural number following 3999 and preceding 4001. It is a decagonal number.

6000 is the natural number following 5999 and preceding 6001.

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

138 is the natural number following 137 and preceding 139.

168 is the natural number following 167 and preceding 169.

744 is the natural number following 743 and preceding 745.

888 is the natural number following 887 and preceding 889.

References

  1. 1 2 Sloane, N. J. A. (ed.). "SequenceA001106(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.). "SequenceA082897(Perfect totient numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 26 May 2016.
  3. 1 2 3 Sloane, N. J. A. (ed.). "SequenceA006872(Numbers k such that phi(k) is phi(sigma(k)).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 3 February 2024.
  4. 1 2 3 Sloane, N. J. A. (ed.). "SequenceA002808(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.
  6. Sloane, N. J. A. (ed.). "SequenceA006003(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.
  8. Sloane, N. J. A. (ed.). "SequenceA002275(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.). "SequenceA000787(Strobogrammatic numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 7 May 2022.
  10. Sloane, N. J. A. (ed.). "SequenceA005349(Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 26 May 2016.
  11. 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